LMFDB - Embedded newform 882.6.a.k.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [882,6,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, 6, names="a")

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

chi := DirichletCharacter("882.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 882 = 2 \cdot 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
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:	$141.458529075$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 18)
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)

$f(q)$

$=$

$q-4.00000 q^{2} +16.0000 q^{4} +96.0000 q^{5} -64.0000 q^{8} +O(q^{10})$

$q-4.00000 q^{2} +16.0000 q^{4} +96.0000 q^{5} -64.0000 q^{8} -384.000 q^{10} +384.000 q^{11} +334.000 q^{13} +256.000 q^{16} -576.000 q^{17} +664.000 q^{19} +1536.00 q^{20} -1536.00 q^{22} -3840.00 q^{23} +6091.00 q^{25} -1336.00 q^{26} +96.0000 q^{29} +4564.00 q^{31} -1024.00 q^{32} +2304.00 q^{34} +5798.00 q^{37} -2656.00 q^{38} -6144.00 q^{40} +6720.00 q^{41} -14872.0 q^{43} +6144.00 q^{44} +15360.0 q^{46} +19200.0 q^{47} -24364.0 q^{50} +5344.00 q^{52} +7776.00 q^{53} +36864.0 q^{55} -384.000 q^{58} +13056.0 q^{59} -42782.0 q^{61} -18256.0 q^{62} +4096.00 q^{64} +32064.0 q^{65} +36656.0 q^{67} -9216.00 q^{68} +64512.0 q^{71} +16810.0 q^{73} -23192.0 q^{74} +10624.0 q^{76} +28076.0 q^{79} +24576.0 q^{80} -26880.0 q^{82} +66432.0 q^{83} -55296.0 q^{85} +59488.0 q^{86} -24576.0 q^{88} +81792.0 q^{89} -61440.0 q^{92} -76800.0 q^{94} +63744.0 q^{95} +29938.0 q^{97} +O(q^{100})$

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 $

$2$	−4.00000	−0.707107
$2$	−4.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	16.0000	0.500000
$5$	96.0000	1.71730	0.858650	−	0.512562i	$-0.171304\pi$
$5$	96.0000	1.71730	0.858650	+	0.512562i	$0.171304\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−64.0000	−0.353553
$9$	0	0
$10$	−384.000	−1.21431
$11$	384.000	0.956862	0.478431	−	0.878125i	$-0.341206\pi$
$11$	384.000	0.956862	0.478431	+	0.878125i	$0.341206\pi$
$12$	0	0
$13$	334.000	0.548136	0.274068	−	0.961710i	$-0.411631\pi$
$13$	334.000	0.548136	0.274068	+	0.961710i	$0.411631\pi$
$14$	0	0
$15$	0	0
$16$	256.000	0.250000
$17$	−576.000	−0.483393	−0.241696	−	0.970352i	$-0.577704\pi$
$17$	−576.000	−0.483393	−0.241696	+	0.970352i	$0.577704\pi$
$18$	0	0
$19$	664.000	0.421972	0.210986	−	0.977489i	$-0.432332\pi$
$19$	664.000	0.421972	0.210986	+	0.977489i	$0.432332\pi$
$20$	1536.00	0.858650
$21$	0	0
$22$	−1536.00	−0.676604
$23$	−3840.00	−1.51360	−0.756801	−	0.653645i	$-0.773239\pi$
$23$	−3840.00	−1.51360	−0.756801	+	0.653645i	$0.773239\pi$
$24$	0	0
$25$	6091.00	1.94912
$26$	−1336.00	−0.387590
$27$	0	0
$28$	0	0
$29$	96.0000	0.0211971	0.0105985	−	0.999944i	$-0.496626\pi$
$29$	96.0000	0.0211971	0.0105985	+	0.999944i	$0.496626\pi$
$30$	0	0
$31$	4564.00	0.852985	0.426493	−	0.904491i	$-0.359749\pi$
$31$	4564.00	0.852985	0.426493	+	0.904491i	$0.359749\pi$
$32$	−1024.00	−0.176777
$33$	0	0
$34$	2304.00	0.341810
$35$	0	0
$36$	0	0
$37$	5798.00	0.696264	0.348132	−	0.937446i	$-0.386816\pi$
$37$	5798.00	0.696264	0.348132	+	0.937446i	$0.386816\pi$
$38$	−2656.00	−0.298380
$39$	0	0
$40$	−6144.00	−0.607157
$41$	6720.00	0.624323	0.312162	−	0.950029i	$-0.398947\pi$
$41$	6720.00	0.624323	0.312162	+	0.950029i	$0.398947\pi$
$42$	0	0
$43$	−14872.0	−1.22659	−0.613293	−	0.789855i	$-0.710156\pi$
$43$	−14872.0	−1.22659	−0.613293	+	0.789855i	$0.710156\pi$
$44$	6144.00	0.478431
$45$	0	0
$46$	15360.0	1.07028
$47$	19200.0	1.26782	0.633909	−	0.773408i	$-0.281450\pi$
$47$	19200.0	1.26782	0.633909	+	0.773408i	$0.281450\pi$
$48$	0	0
$49$	0	0
$50$	−24364.0	−1.37824
$51$	0	0
$52$	5344.00	0.274068
$53$	7776.00	0.380248	0.190124	−	0.981760i	$-0.439111\pi$
$53$	7776.00	0.380248	0.190124	+	0.981760i	$0.439111\pi$
$54$	0	0
$55$	36864.0	1.64322
$56$	0	0
$57$	0	0
$58$	−384.000	−0.0149886
$59$	13056.0	0.488293	0.244146	−	0.969738i	$-0.421492\pi$
$59$	13056.0	0.488293	0.244146	+	0.969738i	$0.421492\pi$
$60$	0	0
$61$	−42782.0	−1.47210	−0.736049	−	0.676929i	$-0.763311\pi$
$61$	−42782.0	−1.47210	−0.736049	+	0.676929i	$0.763311\pi$
$62$	−18256.0	−0.603151
$63$	0	0
$64$	4096.00	0.125000
$65$	32064.0	0.941314
$66$	0	0
$67$	36656.0	0.997604	0.498802	−	0.866716i	$-0.333774\pi$
$67$	36656.0	0.997604	0.498802	+	0.866716i	$0.333774\pi$
$68$	−9216.00	−0.241696
$69$	0	0
$70$	0	0
$71$	64512.0	1.51878	0.759390	−	0.650636i	$-0.225498\pi$
$71$	64512.0	1.51878	0.759390	+	0.650636i	$0.225498\pi$
$72$	0	0
$73$	16810.0	0.369199	0.184600	−	0.982814i	$-0.440901\pi$
$73$	16810.0	0.369199	0.184600	+	0.982814i	$0.440901\pi$
$74$	−23192.0	−0.492333
$75$	0	0
$76$	10624.0	0.210986
$77$	0	0
$78$	0	0
$79$	28076.0	0.506136	0.253068	−	0.967448i	$-0.418560\pi$
$79$	28076.0	0.506136	0.253068	+	0.967448i	$0.418560\pi$
$80$	24576.0	0.429325
$81$	0	0
$82$	−26880.0	−0.441463
$83$	66432.0	1.05848	0.529239	−	0.848473i	$-0.322477\pi$
$83$	66432.0	1.05848	0.529239	+	0.848473i	$0.322477\pi$
$84$	0	0
$85$	−55296.0	−0.830131
$86$	59488.0	0.867328
$87$	0	0
$88$	−24576.0	−0.338302
$89$	81792.0	1.09455	0.547275	−	0.836953i	$-0.315665\pi$
$89$	81792.0	1.09455	0.547275	+	0.836953i	$0.315665\pi$
$90$	0	0
$91$	0	0
$92$	−61440.0	−0.756801
$93$	0	0
$94$	−76800.0	−0.896482
$95$	63744.0	0.724653
$96$	0	0
$97$	29938.0	0.323068	0.161534	−	0.986867i	$-0.448356\pi$
$97$	29938.0	0.323068	0.161534	+	0.986867i	$0.448356\pi$
$98$	0	0
$99$	0	0
$100$	97456.0	0.974560
$101$	−178656.	−1.74267	−0.871333	−	0.490692i	$-0.836744\pi$
$101$	−178656.	−1.74267	−0.871333	+	0.490692i	$0.836744\pi$
$102$	0	0
$103$	115228.	1.07020	0.535100	−	0.844789i	$-0.320274\pi$
$103$	115228.	1.07020	0.535100	+	0.844789i	$0.320274\pi$
$104$	−21376.0	−0.193795
$105$	0	0
$106$	−31104.0	−0.268876
$107$	−76032.0	−0.642003	−0.321001	−	0.947079i	$-0.604019\pi$
$107$	−76032.0	−0.642003	−0.321001	+	0.947079i	$0.604019\pi$
$108$	0	0
$109$	−231118.	−1.86323	−0.931617	−	0.363441i	$-0.881602\pi$
$109$	−231118.	−1.86323	−0.931617	+	0.363441i	$0.881602\pi$
$110$	−147456.	−1.16193
$111$	0	0
$112$	0	0
$113$	142464.	1.04956	0.524782	−	0.851237i	$-0.324147\pi$
$113$	142464.	1.04956	0.524782	+	0.851237i	$0.324147\pi$
$114$	0	0
$115$	−368640.	−2.59931
$116$	1536.00	0.0105985
$117$	0	0
$118$	−52224.0	−0.345275
$119$	0	0
$120$	0	0
$121$	−13595.0	−0.0844143
$122$	171128.	1.04093
$123$	0	0
$124$	73024.0	0.426493
$125$	284736.	1.62992
$126$	0	0
$127$	−988.000	−0.00543560	−0.00271780	−	0.999996i	$-0.500865\pi$
$127$	−988.000	−0.00543560	−0.00271780	+	0.999996i	$0.500865\pi$
$128$	−16384.0	−0.0883883
$129$	0	0
$130$	−128256.	−0.665609
$131$	−224256.	−1.14174	−0.570868	−	0.821042i	$-0.693393\pi$
$131$	−224256.	−1.14174	−0.570868	+	0.821042i	$0.693393\pi$
$132$	0	0
$133$	0	0
$134$	−146624.	−0.705412
$135$	0	0
$136$	36864.0	0.170905
$137$	−278976.	−1.26989	−0.634944	−	0.772558i	$-0.718977\pi$
$137$	−278976.	−1.26989	−0.634944	+	0.772558i	$0.718977\pi$
$138$	0	0
$139$	−177200.	−0.777905	−0.388953	−	0.921258i	$-0.627163\pi$
$139$	−177200.	−0.777905	−0.388953	+	0.921258i	$0.627163\pi$
$140$	0	0
$141$	0	0
$142$	−258048.	−1.07394
$143$	128256.	0.524490
$144$	0	0
$145$	9216.00	0.0364018
$146$	−67240.0	−0.261063
$147$	0	0
$148$	92768.0	0.348132
$149$	236064.	0.871092	0.435546	−	0.900166i	$-0.356555\pi$
$149$	236064.	0.871092	0.435546	+	0.900166i	$0.356555\pi$
$150$	0	0
$151$	−482836.	−1.72329	−0.861643	−	0.507515i	$-0.830564\pi$
$151$	−482836.	−1.72329	−0.861643	+	0.507515i	$0.830564\pi$
$152$	−42496.0	−0.149190
$153$	0	0
$154$	0	0
$155$	438144.	1.46483
$156$	0	0
$157$	−381086.	−1.23388	−0.616941	−	0.787009i	$-0.711628\pi$
$157$	−381086.	−1.23388	−0.616941	+	0.787009i	$0.711628\pi$
$158$	−112304.	−0.357892
$159$	0	0
$160$	−98304.0	−0.303579
$161$	0	0
$162$	0	0
$163$	162920.	0.480292	0.240146	−	0.970737i	$-0.422805\pi$
$163$	162920.	0.480292	0.240146	+	0.970737i	$0.422805\pi$
$164$	107520.	0.312162
$165$	0	0
$166$	−265728.	−0.748457
$167$	566016.	1.57050	0.785249	−	0.619180i	$-0.212535\pi$
$167$	566016.	1.57050	0.785249	+	0.619180i	$0.212535\pi$
$168$	0	0
$169$	−259737.	−0.699547
$170$	221184.	0.586991
$171$	0	0
$172$	−237952.	−0.613293
$173$	218208.	0.554313	0.277157	−	0.960825i	$-0.410608\pi$
$173$	218208.	0.554313	0.277157	+	0.960825i	$0.410608\pi$
$174$	0	0
$175$	0	0
$176$	98304.0	0.239216
$177$	0	0
$178$	−327168.	−0.773964
$179$	−412416.	−0.962062	−0.481031	−	0.876704i	$-0.659738\pi$
$179$	−412416.	−0.962062	−0.481031	+	0.876704i	$0.659738\pi$
$180$	0	0
$181$	25558.0	0.0579870	0.0289935	−	0.999580i	$-0.490770\pi$
$181$	25558.0	0.0579870	0.0289935	+	0.999580i	$0.490770\pi$
$182$	0	0
$183$	0	0
$184$	245760.	0.535139
$185$	556608.	1.19569
$186$	0	0
$187$	−221184.	−0.462540
$188$	307200.	0.633909
$189$	0	0
$190$	−254976.	−0.512407
$191$	400128.	0.793625	0.396813	−	0.917900i	$-0.370116\pi$
$191$	400128.	0.793625	0.396813	+	0.917900i	$0.370116\pi$
$192$	0	0
$193$	699650.	1.35203	0.676017	−	0.736886i	$-0.263705\pi$
$193$	699650.	1.35203	0.676017	+	0.736886i	$0.263705\pi$
$194$	−119752.	−0.228443
$195$	0	0
$196$	0	0
$197$	406368.	0.746026	0.373013	−	0.927826i	$-0.378325\pi$
$197$	406368.	0.746026	0.373013	+	0.927826i	$0.378325\pi$
$198$	0	0
$199$	361996.	0.647994	0.323997	−	0.946058i	$-0.394973\pi$
$199$	361996.	0.647994	0.323997	+	0.946058i	$0.394973\pi$
$200$	−389824.	−0.689118
$201$	0	0
$202$	714624.	1.23225
$203$	0	0
$204$	0	0
$205$	645120.	1.07215
$206$	−460912.	−0.756746
$207$	0	0
$208$	85504.0	0.137034
$209$	254976.	0.403770
$210$	0	0
$211$	151856.	0.234815	0.117407	−	0.993084i	$-0.462542\pi$
$211$	151856.	0.234815	0.117407	+	0.993084i	$0.462542\pi$
$212$	124416.	0.190124
$213$	0	0
$214$	304128.	0.453965
$215$	−1.42771e6	−2.10642
$216$	0	0
$217$	0	0
$218$	924472.	1.31751
$219$	0	0
$220$	589824.	0.821610
$221$	−192384.	−0.264965
$222$	0	0
$223$	1.09332e6	1.47227	0.736134	−	0.676836i	$-0.236649\pi$
$223$	1.09332e6	1.47227	0.736134	+	0.676836i	$0.236649\pi$
$224$	0	0
$225$	0	0
$226$	−569856.	−0.742154
$227$	566400.	0.729556	0.364778	−	0.931095i	$-0.381145\pi$
$227$	566400.	0.729556	0.364778	+	0.931095i	$0.381145\pi$
$228$	0	0
$229$	587206.	0.739949	0.369974	−	0.929042i	$-0.379366\pi$
$229$	587206.	0.739949	0.369974	+	0.929042i	$0.379366\pi$
$230$	1.47456e6	1.83799
$231$	0	0
$232$	−6144.00	−0.00749430
$233$	579456.	0.699247	0.349624	−	0.936890i	$-0.386309\pi$
$233$	579456.	0.699247	0.349624	+	0.936890i	$0.386309\pi$
$234$	0	0
$235$	1.84320e6	2.17722
$236$	208896.	0.244146
$237$	0	0
$238$	0	0
$239$	−584448.	−0.661837	−0.330919	−	0.943659i	$-0.607359\pi$
$239$	−584448.	−0.661837	−0.330919	+	0.943659i	$0.607359\pi$
$240$	0	0
$241$	414130.	0.459298	0.229649	−	0.973274i	$-0.426242\pi$
$241$	414130.	0.459298	0.229649	+	0.973274i	$0.426242\pi$
$242$	54380.0	0.0596899
$243$	0	0
$244$	−684512.	−0.736049
$245$	0	0
$246$	0	0
$247$	221776.	0.231298
$248$	−292096.	−0.301576
$249$	0	0
$250$	−1.13894e6	−1.15253
$251$	1.89965e6	1.90322	0.951610	−	0.307309i	$-0.0994287\pi$
$251$	1.89965e6	1.90322	0.951610	+	0.307309i	$0.0994287\pi$
$252$	0	0
$253$	−1.47456e6	−1.44831
$254$	3952.00	0.00384355
$255$	0	0
$256$	65536.0	0.0625000
$257$	−447744.	−0.422860	−0.211430	−	0.977393i	$-0.567812\pi$
$257$	−447744.	−0.422860	−0.211430	+	0.977393i	$0.567812\pi$
$258$	0	0
$259$	0	0
$260$	513024.	0.470657
$261$	0	0
$262$	897024.	0.807330
$263$	−67584.0	−0.0602496	−0.0301248	−	0.999546i	$-0.509590\pi$
$263$	−67584.0	−0.0602496	−0.0301248	+	0.999546i	$0.509590\pi$
$264$	0	0
$265$	746496.	0.652999
$266$	0	0
$267$	0	0
$268$	586496.	0.498802
$269$	−564192.	−0.475386	−0.237693	−	0.971340i	$-0.576391\pi$
$269$	−564192.	−0.475386	−0.237693	+	0.971340i	$0.576391\pi$
$270$	0	0
$271$	−720308.	−0.595792	−0.297896	−	0.954598i	$-0.596285\pi$
$271$	−720308.	−0.595792	−0.297896	+	0.954598i	$0.596285\pi$
$272$	−147456.	−0.120848
$273$	0	0
$274$	1.11590e6	0.897946
$275$	2.33894e6	1.86504
$276$	0	0
$277$	−141142.	−0.110524	−0.0552620	−	0.998472i	$-0.517599\pi$
$277$	−141142.	−0.110524	−0.0552620	+	0.998472i	$0.517599\pi$
$278$	708800.	0.550062
$279$	0	0
$280$	0	0
$281$	584448.	0.441550	0.220775	−	0.975325i	$-0.429141\pi$
$281$	584448.	0.441550	0.220775	+	0.975325i	$0.429141\pi$
$282$	0	0
$283$	−177056.	−0.131415	−0.0657074	−	0.997839i	$-0.520930\pi$
$283$	−177056.	−0.131415	−0.0657074	+	0.997839i	$0.520930\pi$
$284$	1.03219e6	0.759390
$285$	0	0
$286$	−513024.	−0.370871
$287$	0	0
$288$	0	0
$289$	−1.08808e6	−0.766331
$290$	−36864.0	−0.0257399
$291$	0	0
$292$	268960.	0.184600
$293$	956832.	0.651128	0.325564	−	0.945520i	$-0.394446\pi$
$293$	956832.	0.651128	0.325564	+	0.945520i	$0.394446\pi$
$294$	0	0
$295$	1.25338e6	0.838545
$296$	−371072.	−0.246166
$297$	0	0
$298$	−944256.	−0.615955
$299$	−1.28256e6	−0.829659
$300$	0	0
$301$	0	0
$302$	1.93134e6	1.21855
$303$	0	0
$304$	169984.	0.105493
$305$	−4.10707e6	−2.52803
$306$	0	0
$307$	−2.88286e6	−1.74573	−0.872867	−	0.487958i	$-0.837742\pi$
$307$	−2.88286e6	−1.74573	−0.872867	+	0.487958i	$0.837742\pi$
$308$	0	0
$309$	0	0
$310$	−1.75258e6	−1.03579
$311$	2.60045e6	1.52457	0.762285	−	0.647242i	$-0.224078\pi$
$311$	2.60045e6	1.52457	0.762285	+	0.647242i	$0.224078\pi$
$312$	0	0
$313$	2.58079e6	1.48899	0.744495	−	0.667628i	$-0.232690\pi$
$313$	2.58079e6	1.48899	0.744495	+	0.667628i	$0.232690\pi$
$314$	1.52434e6	0.872487
$315$	0	0
$316$	449216.	0.253068
$317$	2.31101e6	1.29168	0.645838	−	0.763475i	$-0.276508\pi$
$317$	2.31101e6	1.29168	0.645838	+	0.763475i	$0.276508\pi$
$318$	0	0
$319$	36864.0	0.0202827
$320$	393216.	0.214663
$321$	0	0
$322$	0	0
$323$	−382464.	−0.203978
$324$	0	0
$325$	2.03439e6	1.06838
$326$	−651680.	−0.339618
$327$	0	0
$328$	−430080.	−0.220732
$329$	0	0
$330$	0	0
$331$	−637024.	−0.319585	−0.159792	−	0.987151i	$-0.551082\pi$
$331$	−637024.	−0.319585	−0.159792	+	0.987151i	$0.551082\pi$
$332$	1.06291e6	0.529239
$333$	0	0
$334$	−2.26406e6	−1.11051
$335$	3.51898e6	1.71319
$336$	0	0
$337$	3.38665e6	1.62441	0.812206	−	0.583371i	$-0.198267\pi$
$337$	3.38665e6	1.62441	0.812206	+	0.583371i	$0.198267\pi$
$338$	1.03895e6	0.494655
$339$	0	0
$340$	−884736.	−0.415065
$341$	1.75258e6	0.816189
$342$	0	0
$343$	0	0
$344$	951808.	0.433664
$345$	0	0
$346$	−872832.	−0.391959
$347$	2.77824e6	1.23864	0.619321	−	0.785138i	$-0.287408\pi$
$347$	2.77824e6	1.23864	0.619321	+	0.785138i	$0.287408\pi$
$348$	0	0
$349$	−1.55536e6	−0.683545	−0.341772	−	0.939783i	$-0.611027\pi$
$349$	−1.55536e6	−0.683545	−0.341772	+	0.939783i	$0.611027\pi$
$350$	0	0
$351$	0	0
$352$	−393216.	−0.169151
$353$	−2.11776e6	−0.904565	−0.452283	−	0.891875i	$-0.649390\pi$
$353$	−2.11776e6	−0.904565	−0.452283	+	0.891875i	$0.649390\pi$
$354$	0	0
$355$	6.19315e6	2.60820
$356$	1.30867e6	0.547275
$357$	0	0
$358$	1.64966e6	0.680280
$359$	2.17498e6	0.890673	0.445337	−	0.895363i	$-0.353084\pi$
$359$	2.17498e6	0.890673	0.445337	+	0.895363i	$0.353084\pi$
$360$	0	0
$361$	−2.03520e6	−0.821939
$362$	−102232.	−0.0410030
$363$	0	0
$364$	0	0
$365$	1.61376e6	0.634026
$366$	0	0
$367$	−1.05336e6	−0.408235	−0.204117	−	0.978946i	$-0.565432\pi$
$367$	−1.05336e6	−0.408235	−0.204117	+	0.978946i	$0.565432\pi$
$368$	−983040.	−0.378400
$369$	0	0
$370$	−2.22643e6	−0.845483
$371$	0	0
$372$	0	0
$373$	−677098.	−0.251988	−0.125994	−	0.992031i	$-0.540212\pi$
$373$	−677098.	−0.251988	−0.125994	+	0.992031i	$0.540212\pi$
$374$	884736.	0.327065
$375$	0	0
$376$	−1.22880e6	−0.448241
$377$	32064.0	0.0116189
$378$	0	0
$379$	−5.10748e6	−1.82645	−0.913227	−	0.407452i	$-0.866418\pi$
$379$	−5.10748e6	−1.82645	−0.913227	+	0.407452i	$0.866418\pi$
$380$	1.01990e6	0.362327
$381$	0	0
$382$	−1.60051e6	−0.561178
$383$	−1.63200e6	−0.568491	−0.284245	−	0.958752i	$-0.591743\pi$
$383$	−1.63200e6	−0.568491	−0.284245	+	0.958752i	$0.591743\pi$
$384$	0	0
$385$	0	0
$386$	−2.79860e6	−0.956032
$387$	0	0
$388$	479008.	0.161534
$389$	−4.46563e6	−1.49627	−0.748133	−	0.663549i	$-0.769050\pi$
$389$	−4.46563e6	−1.49627	−0.748133	+	0.663549i	$0.769050\pi$
$390$	0	0
$391$	2.21184e6	0.731664
$392$	0	0
$393$	0	0
$394$	−1.62547e6	−0.527520
$395$	2.69530e6	0.869188
$396$	0	0
$397$	611026.	0.194573	0.0972867	−	0.995256i	$-0.468984\pi$
$397$	611026.	0.194573	0.0972867	+	0.995256i	$0.468984\pi$
$398$	−1.44798e6	−0.458201
$399$	0	0
$400$	1.55930e6	0.487280
$401$	−6.09158e6	−1.89177	−0.945887	−	0.324496i	$-0.894805\pi$
$401$	−6.09158e6	−1.89177	−0.945887	+	0.324496i	$0.894805\pi$
$402$	0	0
$403$	1.52438e6	0.467552
$404$	−2.85850e6	−0.871333
$405$	0	0
$406$	0	0
$407$	2.22643e6	0.666229
$408$	0	0
$409$	−2.89108e6	−0.854578	−0.427289	−	0.904115i	$-0.640531\pi$
$409$	−2.89108e6	−0.854578	−0.427289	+	0.904115i	$0.640531\pi$
$410$	−2.58048e6	−0.758125
$411$	0	0
$412$	1.84365e6	0.535100
$413$	0	0
$414$	0	0
$415$	6.37747e6	1.81773
$416$	−342016.	−0.0968976
$417$	0	0
$418$	−1.01990e6	−0.285508
$419$	2.98406e6	0.830373	0.415186	−	0.909736i	$-0.363716\pi$
$419$	2.98406e6	0.830373	0.415186	+	0.909736i	$0.363716\pi$
$420$	0	0
$421$	822074.	0.226051	0.113025	−	0.993592i	$-0.463946\pi$
$421$	822074.	0.226051	0.113025	+	0.993592i	$0.463946\pi$
$422$	−607424.	−0.166039
$423$	0	0
$424$	−497664.	−0.134438
$425$	−3.50842e6	−0.942191
$426$	0	0
$427$	0	0
$428$	−1.21651e6	−0.321001
$429$	0	0
$430$	5.71085e6	1.48946
$431$	6.32448e6	1.63995	0.819977	−	0.572397i	$-0.193986\pi$
$431$	6.32448e6	1.63995	0.819977	+	0.572397i	$0.193986\pi$
$432$	0	0
$433$	851902.	0.218358	0.109179	−	0.994022i	$-0.465178\pi$
$433$	851902.	0.218358	0.109179	+	0.994022i	$0.465178\pi$
$434$	0	0
$435$	0	0
$436$	−3.69789e6	−0.931617
$437$	−2.54976e6	−0.638698
$438$	0	0
$439$	334732.	0.0828964	0.0414482	−	0.999141i	$-0.486803\pi$
$439$	334732.	0.0828964	0.0414482	+	0.999141i	$0.486803\pi$
$440$	−2.35930e6	−0.580966
$441$	0	0
$442$	769536.	0.187358
$443$	−1.76218e6	−0.426619	−0.213309	−	0.976985i	$-0.568424\pi$
$443$	−1.76218e6	−0.426619	−0.213309	+	0.976985i	$0.568424\pi$
$444$	0	0
$445$	7.85203e6	1.87967
$446$	−4.37330e6	−1.04105
$447$	0	0
$448$	0	0
$449$	6.51859e6	1.52594	0.762971	−	0.646433i	$-0.223740\pi$
$449$	6.51859e6	1.52594	0.762971	+	0.646433i	$0.223740\pi$
$450$	0	0
$451$	2.58048e6	0.597392
$452$	2.27942e6	0.524782
$453$	0	0
$454$	−2.26560e6	−0.515874
$455$	0	0
$456$	0	0
$457$	−92074.0	−0.0206227	−0.0103114	−	0.999947i	$-0.503282\pi$
$457$	−92074.0	−0.0206227	−0.0103114	+	0.999947i	$0.503282\pi$
$458$	−2.34882e6	−0.523223
$459$	0	0
$460$	−5.89824e6	−1.29965
$461$	−257568.	−0.0564468	−0.0282234	−	0.999602i	$-0.508985\pi$
$461$	−257568.	−0.0564468	−0.0282234	+	0.999602i	$0.508985\pi$
$462$	0	0
$463$	3.96228e6	0.858998	0.429499	−	0.903067i	$-0.358690\pi$
$463$	3.96228e6	0.858998	0.429499	+	0.903067i	$0.358690\pi$
$464$	24576.0	0.00529927
$465$	0	0
$466$	−2.31782e6	−0.494442
$467$	3.48941e6	0.740388	0.370194	−	0.928954i	$-0.379291\pi$
$467$	3.48941e6	0.740388	0.370194	+	0.928954i	$0.379291\pi$
$468$	0	0
$469$	0	0
$470$	−7.37280e6	−1.53953
$471$	0	0
$472$	−835584.	−0.172637
$473$	−5.71085e6	−1.17367
$474$	0	0
$475$	4.04442e6	0.822475
$476$	0	0
$477$	0	0
$478$	2.33779e6	0.467990
$479$	−513024.	−0.102164	−0.0510821	−	0.998694i	$-0.516267\pi$
$479$	−513024.	−0.102164	−0.0510821	+	0.998694i	$0.516267\pi$
$480$	0	0
$481$	1.93653e6	0.381647
$482$	−1.65652e6	−0.324772
$483$	0	0
$484$	−217520.	−0.0422071
$485$	2.87405e6	0.554804
$486$	0	0
$487$	4.14499e6	0.791956	0.395978	−	0.918260i	$-0.370406\pi$
$487$	4.14499e6	0.791956	0.395978	+	0.918260i	$0.370406\pi$
$488$	2.73805e6	0.520465
$489$	0	0
$490$	0	0
$491$	2.75866e6	0.516409	0.258205	−	0.966090i	$-0.416869\pi$
$491$	2.75866e6	0.516409	0.258205	+	0.966090i	$0.416869\pi$
$492$	0	0
$493$	−55296.0	−0.0102465
$494$	−887104.	−0.163552
$495$	0	0
$496$	1.16838e6	0.213246
$497$	0	0
$498$	0	0
$499$	660896.	0.118818	0.0594089	−	0.998234i	$-0.481078\pi$
$499$	660896.	0.118818	0.0594089	+	0.998234i	$0.481078\pi$
$500$	4.55578e6	0.814962
$501$	0	0
$502$	−7.59859e6	−1.34578
$503$	944640.	0.166474	0.0832370	−	0.996530i	$-0.473474\pi$
$503$	944640.	0.166474	0.0832370	+	0.996530i	$0.473474\pi$
$504$	0	0
$505$	−1.71510e7	−2.99268
$506$	5.89824e6	1.02411
$507$	0	0
$508$	−15808.0	−0.00271780
$509$	7.83773e6	1.34090	0.670449	−	0.741956i	$-0.266102\pi$
$509$	7.83773e6	1.34090	0.670449	+	0.741956i	$0.266102\pi$
$510$	0	0
$511$	0	0
$512$	−262144.	−0.0441942
$513$	0	0
$514$	1.79098e6	0.299007
$515$	1.10619e7	1.83785
$516$	0	0
$517$	7.37280e6	1.21313
$518$	0	0
$519$	0	0
$520$	−2.05210e6	−0.332805
$521$	−3.29645e6	−0.532049	−0.266025	−	0.963966i	$-0.585710\pi$
$521$	−3.29645e6	−0.532049	−0.266025	+	0.963966i	$0.585710\pi$
$522$	0	0
$523$	−6.50238e6	−1.03948	−0.519742	−	0.854323i	$-0.673972\pi$
$523$	−6.50238e6	−1.03948	−0.519742	+	0.854323i	$0.673972\pi$
$524$	−3.58810e6	−0.570868
$525$	0	0
$526$	270336.	0.0426029
$527$	−2.62886e6	−0.412327
$528$	0	0
$529$	8.30926e6	1.29099
$530$	−2.98598e6	−0.461740
$531$	0	0
$532$	0	0
$533$	2.24448e6	0.342214
$534$	0	0
$535$	−7.29907e6	−1.10251
$536$	−2.34598e6	−0.352706
$537$	0	0
$538$	2.25677e6	0.336149
$539$	0	0
$540$	0	0
$541$	9.82714e6	1.44356	0.721778	−	0.692124i	$-0.243325\pi$
$541$	9.82714e6	1.44356	0.721778	+	0.692124i	$0.243325\pi$
$542$	2.88123e6	0.421289
$543$	0	0
$544$	589824.	0.0854526
$545$	−2.21873e7	−3.19973
$546$	0	0
$547$	−3.42580e6	−0.489546	−0.244773	−	0.969580i	$-0.578713\pi$
$547$	−3.42580e6	−0.489546	−0.244773	+	0.969580i	$0.578713\pi$
$548$	−4.46362e6	−0.634944
$549$	0	0
$550$	−9.35578e6	−1.31878
$551$	63744.0	0.00894459
$552$	0	0
$553$	0	0
$554$	564568.	0.0781523
$555$	0	0
$556$	−2.83520e6	−0.388953
$557$	−6.43363e6	−0.878655	−0.439327	−	0.898327i	$-0.644783\pi$
$557$	−6.43363e6	−0.878655	−0.439327	+	0.898327i	$0.644783\pi$
$558$	0	0
$559$	−4.96725e6	−0.672336
$560$	0	0
$561$	0	0
$562$	−2.33779e6	−0.312223
$563$	−753024.	−0.100124	−0.0500620	−	0.998746i	$-0.515942\pi$
$563$	−753024.	−0.100124	−0.0500620	+	0.998746i	$0.515942\pi$
$564$	0	0
$565$	1.36765e7	1.80242
$566$	708224.	0.0929244
$567$	0	0
$568$	−4.12877e6	−0.536970
$569$	−1.11481e7	−1.44351	−0.721755	−	0.692148i	$-0.756664\pi$
$569$	−1.11481e7	−1.44351	−0.721755	+	0.692148i	$0.756664\pi$
$570$	0	0
$571$	191024.	0.0245187	0.0122594	−	0.999925i	$-0.496098\pi$
$571$	191024.	0.0245187	0.0122594	+	0.999925i	$0.496098\pi$
$572$	2.05210e6	0.262245
$573$	0	0
$574$	0	0
$575$	−2.33894e7	−2.95019
$576$	0	0
$577$	−1.03722e7	−1.29697	−0.648486	−	0.761227i	$-0.724597\pi$
$577$	−1.03722e7	−1.29697	−0.648486	+	0.761227i	$0.724597\pi$
$578$	4.35232e6	0.541878
$579$	0	0
$580$	147456.	0.0182009
$581$	0	0
$582$	0	0
$583$	2.98598e6	0.363845
$584$	−1.07584e6	−0.130532
$585$	0	0
$586$	−3.82733e6	−0.460417
$587$	2.97062e6	0.355838	0.177919	−	0.984045i	$-0.443063\pi$
$587$	2.97062e6	0.355838	0.177919	+	0.984045i	$0.443063\pi$
$588$	0	0
$589$	3.03050e6	0.359936
$590$	−5.01350e6	−0.592941
$591$	0	0
$592$	1.48429e6	0.174066
$593$	7.55827e6	0.882644	0.441322	−	0.897349i	$-0.354510\pi$
$593$	7.55827e6	0.882644	0.441322	+	0.897349i	$0.354510\pi$
$594$	0	0
$595$	0	0
$596$	3.77702e6	0.435546
$597$	0	0
$598$	5.13024e6	0.586658
$599$	6.69158e6	0.762012	0.381006	−	0.924573i	$-0.375578\pi$
$599$	6.69158e6	0.762012	0.381006	+	0.924573i	$0.375578\pi$
$600$	0	0
$601$	3.20359e6	0.361785	0.180893	−	0.983503i	$-0.442101\pi$
$601$	3.20359e6	0.361785	0.180893	+	0.983503i	$0.442101\pi$
$602$	0	0
$603$	0	0
$604$	−7.72538e6	−0.861643
$605$	−1.30512e6	−0.144965
$606$	0	0
$607$	1.35585e7	1.49362	0.746809	−	0.665038i	$-0.231585\pi$
$607$	1.35585e7	1.49362	0.746809	+	0.665038i	$0.231585\pi$
$608$	−679936.	−0.0745949
$609$	0	0
$610$	1.64283e7	1.78759
$611$	6.41280e6	0.694936
$612$	0	0
$613$	−1.07654e7	−1.15712	−0.578561	−	0.815639i	$-0.696385\pi$
$613$	−1.07654e7	−1.15712	−0.578561	+	0.815639i	$0.696385\pi$
$614$	1.15315e7	1.23442
$615$	0	0
$616$	0	0
$617$	−9.33504e6	−0.987196	−0.493598	−	0.869690i	$-0.664319\pi$
$617$	−9.33504e6	−0.987196	−0.493598	+	0.869690i	$0.664319\pi$
$618$	0	0
$619$	−9.07664e6	−0.952135	−0.476067	−	0.879409i	$-0.657938\pi$
$619$	−9.07664e6	−0.952135	−0.476067	+	0.879409i	$0.657938\pi$
$620$	7.01030e6	0.732416
$621$	0	0
$622$	−1.04018e7	−1.07803
$623$	0	0
$624$	0	0
$625$	8.30028e6	0.849949
$626$	−1.03232e7	−1.05288
$627$	0	0
$628$	−6.09738e6	−0.616941
$629$	−3.33965e6	−0.336569
$630$	0	0
$631$	−1.13367e7	−1.13348	−0.566741	−	0.823896i	$-0.691796\pi$
$631$	−1.13367e7	−1.13348	−0.566741	+	0.823896i	$0.691796\pi$
$632$	−1.79686e6	−0.178946
$633$	0	0
$634$	−9.24403e6	−0.913352
$635$	−94848.0	−0.00933456
$636$	0	0
$637$	0	0
$638$	−147456.	−0.0143420
$639$	0	0
$640$	−1.57286e6	−0.151789
$641$	1.55449e7	1.49432	0.747159	−	0.664646i	$-0.231418\pi$
$641$	1.55449e7	1.49432	0.747159	+	0.664646i	$0.231418\pi$
$642$	0	0
$643$	8.80026e6	0.839399	0.419699	−	0.907663i	$-0.362136\pi$
$643$	8.80026e6	0.839399	0.419699	+	0.907663i	$0.362136\pi$
$644$	0	0
$645$	0	0
$646$	1.52986e6	0.144235
$647$	−1.08449e7	−1.01851	−0.509256	−	0.860615i	$-0.670079\pi$
$647$	−1.08449e7	−1.01851	−0.509256	+	0.860615i	$0.670079\pi$
$648$	0	0
$649$	5.01350e6	0.467229
$650$	−8.13758e6	−0.755460
$651$	0	0
$652$	2.60672e6	0.240146
$653$	−9.88771e6	−0.907429	−0.453715	−	0.891147i	$-0.649901\pi$
$653$	−9.88771e6	−0.907429	−0.453715	+	0.891147i	$0.649901\pi$
$654$	0	0
$655$	−2.15286e7	−1.96070
$656$	1.72032e6	0.156081
$657$	0	0
$658$	0	0
$659$	−1.46150e7	−1.31095	−0.655476	−	0.755216i	$-0.727532\pi$
$659$	−1.46150e7	−1.31095	−0.655476	+	0.755216i	$0.727532\pi$
$660$	0	0
$661$	−1.57792e7	−1.40469	−0.702347	−	0.711834i	$-0.747865\pi$
$661$	−1.57792e7	−1.40469	−0.702347	+	0.711834i	$0.747865\pi$
$662$	2.54810e6	0.225980
$663$	0	0
$664$	−4.25165e6	−0.374229
$665$	0	0
$666$	0	0
$667$	−368640.	−0.0320840
$668$	9.05626e6	0.785249
$669$	0	0
$670$	−1.40759e7	−1.21140
$671$	−1.64283e7	−1.40859
$672$	0	0
$673$	6.64939e6	0.565906	0.282953	−	0.959134i	$-0.408686\pi$
$673$	6.64939e6	0.565906	0.282953	+	0.959134i	$0.408686\pi$
$674$	−1.35466e7	−1.14863
$675$	0	0
$676$	−4.15579e6	−0.349774
$677$	−4.42550e6	−0.371100	−0.185550	−	0.982635i	$-0.559407\pi$
$677$	−4.42550e6	−0.371100	−0.185550	+	0.982635i	$0.559407\pi$
$678$	0	0
$679$	0	0
$680$	3.53894e6	0.293495
$681$	0	0
$682$	−7.01030e6	−0.577133
$683$	−4.67827e6	−0.383737	−0.191869	−	0.981421i	$-0.561455\pi$
$683$	−4.67827e6	−0.383737	−0.191869	+	0.981421i	$0.561455\pi$
$684$	0	0
$685$	−2.67817e7	−2.18078
$686$	0	0
$687$	0	0
$688$	−3.80723e6	−0.306647
$689$	2.59718e6	0.208427
$690$	0	0
$691$	5.54102e6	0.441463	0.220731	−	0.975335i	$-0.429156\pi$
$691$	5.54102e6	0.441463	0.220731	+	0.975335i	$0.429156\pi$
$692$	3.49133e6	0.277157
$693$	0	0
$694$	−1.11130e7	−0.875853
$695$	−1.70112e7	−1.33590
$696$	0	0
$697$	−3.87072e6	−0.301793
$698$	6.22143e6	0.483339
$699$	0	0
$700$	0	0
$701$	−6.53443e6	−0.502242	−0.251121	−	0.967956i	$-0.580799\pi$
$701$	−6.53443e6	−0.502242	−0.251121	+	0.967956i	$0.580799\pi$
$702$	0	0
$703$	3.84987e6	0.293804
$704$	1.57286e6	0.119608
$705$	0	0
$706$	8.47104e6	0.639624
$707$	0	0
$708$	0	0
$709$	−3.86541e6	−0.288789	−0.144394	−	0.989520i	$-0.546123\pi$
$709$	−3.86541e6	−0.288789	−0.144394	+	0.989520i	$0.546123\pi$
$710$	−2.47726e7	−1.84428
$711$	0	0
$712$	−5.23469e6	−0.386982
$713$	−1.75258e7	−1.29108
$714$	0	0
$715$	1.23126e7	0.900708
$716$	−6.59866e6	−0.481031
$717$	0	0
$718$	−8.69990e6	−0.629801
$719$	−4.80614e6	−0.346717	−0.173358	−	0.984859i	$-0.555462\pi$
$719$	−4.80614e6	−0.346717	−0.173358	+	0.984859i	$0.555462\pi$
$720$	0	0
$721$	0	0
$722$	8.14081e6	0.581199
$723$	0	0
$724$	408928.	0.0289935
$725$	584736.	0.0413157
$726$	0	0
$727$	1.90590e7	1.33741	0.668704	−	0.743529i	$-0.266850\pi$
$727$	1.90590e7	1.33741	0.668704	+	0.743529i	$0.266850\pi$
$728$	0	0
$729$	0	0
$730$	−6.45504e6	−0.448324
$731$	8.56627e6	0.592923
$732$	0	0
$733$	−5.69616e6	−0.391582	−0.195791	−	0.980646i	$-0.562727\pi$
$733$	−5.69616e6	−0.391582	−0.195791	+	0.980646i	$0.562727\pi$
$734$	4.21342e6	0.288666
$735$	0	0
$736$	3.93216e6	0.267570
$737$	1.40759e7	0.954570
$738$	0	0
$739$	1.84902e7	1.24546	0.622730	−	0.782437i	$-0.286024\pi$
$739$	1.84902e7	1.24546	0.622730	+	0.782437i	$0.286024\pi$
$740$	8.90573e6	0.597847
$741$	0	0
$742$	0	0
$743$	9.90336e6	0.658128	0.329064	−	0.944308i	$-0.393267\pi$
$743$	9.90336e6	0.658128	0.329064	+	0.944308i	$0.393267\pi$
$744$	0	0
$745$	2.26621e7	1.49593
$746$	2.70839e6	0.178182
$747$	0	0
$748$	−3.53894e6	−0.231270
$749$	0	0
$750$	0	0
$751$	−9.05914e6	−0.586121	−0.293060	−	0.956094i	$-0.594674\pi$
$751$	−9.05914e6	−0.586121	−0.293060	+	0.956094i	$0.594674\pi$
$752$	4.91520e6	0.316954
$753$	0	0
$754$	−128256.	−0.00821579
$755$	−4.63523e7	−2.95940
$756$	0	0
$757$	−1.16677e7	−0.740022	−0.370011	−	0.929027i	$-0.620646\pi$
$757$	−1.16677e7	−0.740022	−0.370011	+	0.929027i	$0.620646\pi$
$758$	2.04299e7	1.29150
$759$	0	0
$760$	−4.07962e6	−0.256204
$761$	1.27398e7	0.797444	0.398722	−	0.917072i	$-0.369454\pi$
$761$	1.27398e7	0.797444	0.398722	+	0.917072i	$0.369454\pi$
$762$	0	0
$763$	0	0
$764$	6.40205e6	0.396813
$765$	0	0
$766$	6.52800e6	0.401983
$767$	4.36070e6	0.267651
$768$	0	0
$769$	−1.06783e7	−0.651156	−0.325578	−	0.945515i	$-0.605559\pi$
$769$	−1.06783e7	−0.651156	−0.325578	+	0.945515i	$0.605559\pi$
$770$	0	0
$771$	0	0
$772$	1.11944e7	0.676017
$773$	9.18634e6	0.552960	0.276480	−	0.961020i	$-0.410832\pi$
$773$	9.18634e6	0.552960	0.276480	+	0.961020i	$0.410832\pi$
$774$	0	0
$775$	2.77993e7	1.66257
$776$	−1.91603e6	−0.114222
$777$	0	0
$778$	1.78625e7	1.05802
$779$	4.46208e6	0.263447
$780$	0	0
$781$	2.47726e7	1.45326
$782$	−8.84736e6	−0.517365
$783$	0	0
$784$	0	0
$785$	−3.65843e7	−2.11895
$786$	0	0
$787$	9.28209e6	0.534206	0.267103	−	0.963668i	$-0.413934\pi$
$787$	9.28209e6	0.534206	0.267103	+	0.963668i	$0.413934\pi$
$788$	6.50189e6	0.373013
$789$	0	0
$790$	−1.07812e7	−0.614609
$791$	0	0
$792$	0	0
$793$	−1.42892e7	−0.806909
$794$	−2.44410e6	−0.137584
$795$	0	0
$796$	5.79194e6	0.323997
$797$	−21792.0	−0.00121521	−0.000607605	−	1.00000i	$-0.500193\pi$
$797$	−21792.0	−0.00121521	−0.000607605		1.00000i	$0.500193\pi$
$798$	0	0
$799$	−1.10592e7	−0.612854
$800$	−6.23718e6	−0.344559
$801$	0	0
$802$	2.43663e7	1.33769
$803$	6.45504e6	0.353273
$804$	0	0
$805$	0	0
$806$	−6.09750e6	−0.330609
$807$	0	0
$808$	1.14340e7	0.616125
$809$	1.23085e7	0.661204	0.330602	−	0.943770i	$-0.392748\pi$
$809$	1.23085e7	0.661204	0.330602	+	0.943770i	$0.392748\pi$
$810$	0	0
$811$	2.34636e7	1.25269	0.626343	−	0.779547i	$-0.284551\pi$
$811$	2.34636e7	1.25269	0.626343	+	0.779547i	$0.284551\pi$
$812$	0	0
$813$	0	0
$814$	−8.90573e6	−0.471095
$815$	1.56403e7	0.824806
$816$	0	0
$817$	−9.87501e6	−0.517586
$818$	1.15643e7	0.604278
$819$	0	0
$820$	1.03219e7	0.536075
$821$	−1.44206e7	−0.746666	−0.373333	−	0.927697i	$-0.621785\pi$
$821$	−1.44206e7	−0.746666	−0.373333	+	0.927697i	$0.621785\pi$
$822$	0	0
$823$	3.43419e7	1.76736	0.883679	−	0.468093i	$-0.155059\pi$
$823$	3.43419e7	1.76736	0.883679	+	0.468093i	$0.155059\pi$
$824$	−7.37459e6	−0.378373
$825$	0	0
$826$	0	0
$827$	2.13327e7	1.08463	0.542316	−	0.840174i	$-0.317547\pi$
$827$	2.13327e7	1.08463	0.542316	+	0.840174i	$0.317547\pi$
$828$	0	0
$829$	−2.63751e6	−0.133293	−0.0666465	−	0.997777i	$-0.521230\pi$
$829$	−2.63751e6	−0.133293	−0.0666465	+	0.997777i	$0.521230\pi$
$830$	−2.55099e7	−1.28533
$831$	0	0
$832$	1.36806e6	0.0685170
$833$	0	0
$834$	0	0
$835$	5.43375e7	2.69702
$836$	4.07962e6	0.201885
$837$	0	0
$838$	−1.19363e7	−0.587162
$839$	1.00577e7	0.493282	0.246641	−	0.969107i	$-0.420673\pi$
$839$	1.00577e7	0.493282	0.246641	+	0.969107i	$0.420673\pi$
$840$	0	0
$841$	−2.05019e7	−0.999551
$842$	−3.28830e6	−0.159842
$843$	0	0
$844$	2.42970e6	0.117407
$845$	−2.49348e7	−1.20133
$846$	0	0
$847$	0	0
$848$	1.99066e6	0.0950619
$849$	0	0
$850$	1.40337e7	0.666229
$851$	−2.22643e7	−1.05387
$852$	0	0
$853$	2.30748e7	1.08584	0.542919	−	0.839785i	$-0.317319\pi$
$853$	2.30748e7	1.08584	0.542919	+	0.839785i	$0.317319\pi$
$854$	0	0
$855$	0	0
$856$	4.86605e6	0.226982
$857$	−3.51646e7	−1.63551	−0.817756	−	0.575565i	$-0.804782\pi$
$857$	−3.51646e7	−1.63551	−0.817756	+	0.575565i	$0.804782\pi$
$858$	0	0
$859$	−1.66022e7	−0.767684	−0.383842	−	0.923399i	$-0.625399\pi$
$859$	−1.66022e7	−0.767684	−0.383842	+	0.923399i	$0.625399\pi$
$860$	−2.28434e7	−1.05321
$861$	0	0
$862$	−2.52979e7	−1.15962
$863$	−2.97009e7	−1.35751	−0.678754	−	0.734366i	$-0.737480\pi$
$863$	−2.97009e7	−1.35751	−0.678754	+	0.734366i	$0.737480\pi$
$864$	0	0
$865$	2.09480e7	0.951923
$866$	−3.40761e6	−0.154403
$867$	0	0
$868$	0	0
$869$	1.07812e7	0.484303
$870$	0	0
$871$	1.22431e7	0.546822
$872$	1.47916e7	0.658753
$873$	0	0
$874$	1.01990e7	0.451628
$875$	0	0
$876$	0	0
$877$	−1.17943e7	−0.517811	−0.258906	−	0.965903i	$-0.583362\pi$
$877$	−1.17943e7	−0.517811	−0.258906	+	0.965903i	$0.583362\pi$
$878$	−1.33893e6	−0.0586166
$879$	0	0
$880$	9.43718e6	0.410805
$881$	2.10378e7	0.913190	0.456595	−	0.889675i	$-0.349069\pi$
$881$	2.10378e7	0.913190	0.456595	+	0.889675i	$0.349069\pi$
$882$	0	0
$883$	2.12192e7	0.915855	0.457928	−	0.888990i	$-0.348592\pi$
$883$	2.12192e7	0.915855	0.457928	+	0.888990i	$0.348592\pi$
$884$	−3.07814e6	−0.132482
$885$	0	0
$886$	7.04870e6	0.301665
$887$	−2.28818e7	−0.976520	−0.488260	−	0.872698i	$-0.662368\pi$
$887$	−2.28818e7	−0.976520	−0.488260	+	0.872698i	$0.662368\pi$
$888$	0	0
$889$	0	0
$890$	−3.14081e7	−1.32913
$891$	0	0
$892$	1.74932e7	0.736134
$893$	1.27488e7	0.534984
$894$	0	0
$895$	−3.95919e7	−1.65215
$896$	0	0
$897$	0	0
$898$	−2.60744e7	−1.07900
$899$	438144.	0.0180808
$900$	0	0
$901$	−4.47898e6	−0.183809
$902$	−1.03219e7	−0.422420
$903$	0	0
$904$	−9.11770e6	−0.371077
$905$	2.45357e6	0.0995810
$906$	0	0
$907$	−9.68692e6	−0.390992	−0.195496	−	0.980705i	$-0.562632\pi$
$907$	−9.68692e6	−0.390992	−0.195496	+	0.980705i	$0.562632\pi$
$908$	9.06240e6	0.364778
$909$	0	0
$910$	0	0
$911$	9.38112e6	0.374506	0.187253	−	0.982312i	$-0.440042\pi$
$911$	9.38112e6	0.374506	0.187253	+	0.982312i	$0.440042\pi$
$912$	0	0
$913$	2.55099e7	1.01282
$914$	368296.	0.0145825
$915$	0	0
$916$	9.39530e6	0.369974
$917$	0	0
$918$	0	0
$919$	−4.21870e7	−1.64774	−0.823872	−	0.566775i	$-0.808191\pi$
$919$	−4.21870e7	−1.64774	−0.823872	+	0.566775i	$0.808191\pi$
$920$	2.35930e7	0.918994
$921$	0	0
$922$	1.03027e6	0.0399139
$923$	2.15470e7	0.832497
$924$	0	0
$925$	3.53156e7	1.35710
$926$	−1.58491e7	−0.607403
$927$	0	0
$928$	−98304.0	−0.00374715
$929$	3.04556e7	1.15779	0.578893	−	0.815404i	$-0.303485\pi$
$929$	3.04556e7	1.15779	0.578893	+	0.815404i	$0.303485\pi$
$930$	0	0
$931$	0	0
$932$	9.27130e6	0.349624
$933$	0	0
$934$	−1.39576e7	−0.523534
$935$	−2.12337e7	−0.794321
$936$	0	0
$937$	−1.47847e7	−0.550128	−0.275064	−	0.961426i	$-0.588699\pi$
$937$	−1.47847e7	−0.550128	−0.275064	+	0.961426i	$0.588699\pi$
$938$	0	0
$939$	0	0
$940$	2.94912e7	1.08861
$941$	9.91997e6	0.365205	0.182602	−	0.983187i	$-0.441548\pi$
$941$	9.91997e6	0.365205	0.182602	+	0.983187i	$0.441548\pi$
$942$	0	0
$943$	−2.58048e7	−0.944977
$944$	3.34234e6	0.122073
$945$	0	0
$946$	2.28434e7	0.829913
$947$	2.91610e6	0.105664	0.0528320	−	0.998603i	$-0.483175\pi$
$947$	2.91610e6	0.105664	0.0528320	+	0.998603i	$0.483175\pi$
$948$	0	0
$949$	5.61454e6	0.202371
$950$	−1.61777e7	−0.581578
$951$	0	0
$952$	0	0
$953$	1.40861e7	0.502410	0.251205	−	0.967934i	$-0.419173\pi$
$953$	1.40861e7	0.502410	0.251205	+	0.967934i	$0.419173\pi$
$954$	0	0
$955$	3.84123e7	1.36289
$956$	−9.35117e6	−0.330919
$957$	0	0
$958$	2.05210e6	0.0722410
$959$	0	0
$960$	0	0
$961$	−7.79906e6	−0.272417
$962$	−7.74613e6	−0.269865
$963$	0	0
$964$	6.62608e6	0.229649
$965$	6.71664e7	2.32185
$966$	0	0
$967$	1.51949e7	0.522553	0.261276	−	0.965264i	$-0.415857\pi$
$967$	1.51949e7	0.522553	0.261276	+	0.965264i	$0.415857\pi$
$968$	870080.	0.0298449
$969$	0	0
$970$	−1.14962e7	−0.392306
$971$	5.61220e7	1.91023	0.955113	−	0.296240i	$-0.0957329\pi$
$971$	5.61220e7	1.91023	0.955113	+	0.296240i	$0.0957329\pi$
$972$	0	0
$973$	0	0
$974$	−1.65800e7	−0.559997
$975$	0	0
$976$	−1.09522e7	−0.368024
$977$	3.45625e7	1.15843	0.579214	−	0.815176i	$-0.303360\pi$
$977$	3.45625e7	1.15843	0.579214	+	0.815176i	$0.303360\pi$
$978$	0	0
$979$	3.14081e7	1.04733
$980$	0	0
$981$	0	0
$982$	−1.10346e7	−0.365156
$983$	−5.56385e7	−1.83650	−0.918252	−	0.395997i	$-0.870399\pi$
$983$	−5.56385e7	−1.83650	−0.918252	+	0.395997i	$0.870399\pi$
$984$	0	0
$985$	3.90113e7	1.28115
$986$	221184.	0.00724538
$987$	0	0
$988$	3.54842e6	0.115649
$989$	5.71085e7	1.85656
$990$	0	0
$991$	−3.60028e7	−1.16453	−0.582267	−	0.812998i	$-0.697834\pi$
$991$	−3.60028e7	−1.16453	−0.582267	+	0.812998i	$0.697834\pi$
$992$	−4.67354e6	−0.150788
$993$	0	0
$994$	0	0
$995$	3.47516e7	1.11280
$996$	0	0
$997$	−2.35811e7	−0.751322	−0.375661	−	0.926757i	$-0.622584\pi$
$997$	−2.35811e7	−0.751322	−0.375661	+	0.926757i	$0.622584\pi$
$998$	−2.64358e6	−0.0840169
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	882.6.a.k.1.1		1
3.2	odd	2		882.6.a.l.1.1		1
7.6	odd	2		18.6.a.a.1.1	✓	1
21.20	even	2		18.6.a.c.1.1	yes	1
28.27	even	2		144.6.a.a.1.1		1
35.13	even	4		450.6.c.m.199.2		2
35.27	even	4		450.6.c.m.199.1		2
35.34	odd	2		450.6.a.v.1.1		1
56.13	odd	2		576.6.a.bh.1.1		1
56.27	even	2		576.6.a.bi.1.1		1
63.13	odd	6		162.6.c.l.55.1		2
63.20	even	6		162.6.c.a.109.1		2
63.34	odd	6		162.6.c.l.109.1		2
63.41	even	6		162.6.c.a.55.1		2
84.83	odd	2		144.6.a.l.1.1		1
105.62	odd	4		450.6.c.c.199.2		2
105.83	odd	4		450.6.c.c.199.1		2
105.104	even	2		450.6.a.k.1.1		1
168.83	odd	2		576.6.a.b.1.1		1
168.125	even	2		576.6.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
18.6.a.a.1.1	✓	1	7.6	odd	2
18.6.a.c.1.1	yes	1	21.20	even	2
144.6.a.a.1.1		1	28.27	even	2
144.6.a.l.1.1		1	84.83	odd	2
162.6.c.a.55.1		2	63.41	even	6
162.6.c.a.109.1		2	63.20	even	6
162.6.c.l.55.1		2	63.13	odd	6
162.6.c.l.109.1		2	63.34	odd	6
450.6.a.k.1.1		1	105.104	even	2
450.6.a.v.1.1		1	35.34	odd	2
450.6.c.c.199.1		2	105.83	odd	4
450.6.c.c.199.2		2	105.62	odd	4
450.6.c.m.199.1		2	35.27	even	4
450.6.c.m.199.2		2	35.13	even	4
576.6.a.a.1.1		1	168.125	even	2
576.6.a.b.1.1		1	168.83	odd	2
576.6.a.bh.1.1		1	56.13	odd	2
576.6.a.bi.1.1		1	56.27	even	2
882.6.a.k.1.1		1	1.1	even	1	trivial
882.6.a.l.1.1		1	3.2	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:	\( N \)	\(=\)	\( 882 = 2 \cdot 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	882.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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