# Properties

 Label 882.4.a.w.1.1 Level $882$ Weight $4$ Character 882.1 Self dual yes Analytic conductor $52.040$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

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

## Newform invariants

 Self dual: yes Analytic conductor: $$52.0396846251$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{22})$$ Defining polynomial: $$x^{2} - 22$$ Coefficient ring: $$\Z[a_1, \ldots, a_{5}]$$ Coefficient ring index: $$2$$ Twist minimal: no (minimal twist has level 98) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-4.69042$$ of defining polynomial Character $$\chi$$ $$=$$ 882.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.00000 q^{2} +4.00000 q^{4} -9.38083 q^{5} -8.00000 q^{8} +O(q^{10})$$ $$q-2.00000 q^{2} +4.00000 q^{4} -9.38083 q^{5} -8.00000 q^{8} +18.7617 q^{10} -20.0000 q^{11} +65.6658 q^{13} +16.0000 q^{16} -56.2850 q^{17} +9.38083 q^{19} -37.5233 q^{20} +40.0000 q^{22} -48.0000 q^{23} -37.0000 q^{25} -131.332 q^{26} +166.000 q^{29} -206.378 q^{31} -32.0000 q^{32} +112.570 q^{34} -78.0000 q^{37} -18.7617 q^{38} +75.0467 q^{40} -393.995 q^{41} +436.000 q^{43} -80.0000 q^{44} +96.0000 q^{46} -206.378 q^{47} +74.0000 q^{50} +262.663 q^{52} -62.0000 q^{53} +187.617 q^{55} -332.000 q^{58} +666.039 q^{59} +272.044 q^{61} +412.757 q^{62} +64.0000 q^{64} -616.000 q^{65} +580.000 q^{67} -225.140 q^{68} +544.000 q^{71} -600.373 q^{73} +156.000 q^{74} +37.5233 q^{76} -680.000 q^{79} -150.093 q^{80} +787.990 q^{82} -196.997 q^{83} +528.000 q^{85} -872.000 q^{86} +160.000 q^{88} +1500.93 q^{89} -192.000 q^{92} +412.757 q^{94} -88.0000 q^{95} -656.658 q^{97} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2q - 4q^{2} + 8q^{4} - 16q^{8} + O(q^{10})$$ $$2q - 4q^{2} + 8q^{4} - 16q^{8} - 40q^{11} + 32q^{16} + 80q^{22} - 96q^{23} - 74q^{25} + 332q^{29} - 64q^{32} - 156q^{37} + 872q^{43} - 160q^{44} + 192q^{46} + 148q^{50} - 124q^{53} - 664q^{58} + 128q^{64} - 1232q^{65} + 1160q^{67} + 1088q^{71} + 312q^{74} - 1360q^{79} + 1056q^{85} - 1744q^{86} + 320q^{88} - 384q^{92} - 176q^{95} + O(q^{100})$$

## 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 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.00000 −0.707107
$$3$$ 0 0
$$4$$ 4.00000 0.500000
$$5$$ −9.38083 −0.839047 −0.419524 0.907744i $$-0.637803\pi$$
−0.419524 + 0.907744i $$0.637803\pi$$
$$6$$ 0 0
$$7$$ 0 0
$$8$$ −8.00000 −0.353553
$$9$$ 0 0
$$10$$ 18.7617 0.593296
$$11$$ −20.0000 −0.548202 −0.274101 0.961701i $$-0.588380\pi$$
−0.274101 + 0.961701i $$0.588380\pi$$
$$12$$ 0 0
$$13$$ 65.6658 1.40096 0.700478 0.713674i $$-0.252970\pi$$
0.700478 + 0.713674i $$0.252970\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 16.0000 0.250000
$$17$$ −56.2850 −0.803007 −0.401503 0.915858i $$-0.631512\pi$$
−0.401503 + 0.915858i $$0.631512\pi$$
$$18$$ 0 0
$$19$$ 9.38083 0.113269 0.0566345 0.998395i $$-0.481963\pi$$
0.0566345 + 0.998395i $$0.481963\pi$$
$$20$$ −37.5233 −0.419524
$$21$$ 0 0
$$22$$ 40.0000 0.387638
$$23$$ −48.0000 −0.435161 −0.217580 0.976042i $$-0.569816\pi$$
−0.217580 + 0.976042i $$0.569816\pi$$
$$24$$ 0 0
$$25$$ −37.0000 −0.296000
$$26$$ −131.332 −0.990625
$$27$$ 0 0
$$28$$ 0 0
$$29$$ 166.000 1.06295 0.531473 0.847075i $$-0.321639\pi$$
0.531473 + 0.847075i $$0.321639\pi$$
$$30$$ 0 0
$$31$$ −206.378 −1.19570 −0.597849 0.801609i $$-0.703978\pi$$
−0.597849 + 0.801609i $$0.703978\pi$$
$$32$$ −32.0000 −0.176777
$$33$$ 0 0
$$34$$ 112.570 0.567812
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −78.0000 −0.346571 −0.173285 0.984872i $$-0.555438\pi$$
−0.173285 + 0.984872i $$0.555438\pi$$
$$38$$ −18.7617 −0.0800933
$$39$$ 0 0
$$40$$ 75.0467 0.296648
$$41$$ −393.995 −1.50077 −0.750386 0.661000i $$-0.770132\pi$$
−0.750386 + 0.661000i $$0.770132\pi$$
$$42$$ 0 0
$$43$$ 436.000 1.54626 0.773132 0.634245i $$-0.218689\pi$$
0.773132 + 0.634245i $$0.218689\pi$$
$$44$$ −80.0000 −0.274101
$$45$$ 0 0
$$46$$ 96.0000 0.307705
$$47$$ −206.378 −0.640497 −0.320249 0.947334i $$-0.603766\pi$$
−0.320249 + 0.947334i $$0.603766\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 74.0000 0.209304
$$51$$ 0 0
$$52$$ 262.663 0.700478
$$53$$ −62.0000 −0.160686 −0.0803430 0.996767i $$-0.525602\pi$$
−0.0803430 + 0.996767i $$0.525602\pi$$
$$54$$ 0 0
$$55$$ 187.617 0.459968
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −332.000 −0.751616
$$59$$ 666.039 1.46968 0.734838 0.678243i $$-0.237258\pi$$
0.734838 + 0.678243i $$0.237258\pi$$
$$60$$ 0 0
$$61$$ 272.044 0.571011 0.285506 0.958377i $$-0.407838\pi$$
0.285506 + 0.958377i $$0.407838\pi$$
$$62$$ 412.757 0.845486
$$63$$ 0 0
$$64$$ 64.0000 0.125000
$$65$$ −616.000 −1.17547
$$66$$ 0 0
$$67$$ 580.000 1.05759 0.528793 0.848751i $$-0.322645\pi$$
0.528793 + 0.848751i $$0.322645\pi$$
$$68$$ −225.140 −0.401503
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 544.000 0.909309 0.454654 0.890668i $$-0.349763\pi$$
0.454654 + 0.890668i $$0.349763\pi$$
$$72$$ 0 0
$$73$$ −600.373 −0.962580 −0.481290 0.876561i $$-0.659832\pi$$
−0.481290 + 0.876561i $$0.659832\pi$$
$$74$$ 156.000 0.245063
$$75$$ 0 0
$$76$$ 37.5233 0.0566345
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −680.000 −0.968430 −0.484215 0.874949i $$-0.660895\pi$$
−0.484215 + 0.874949i $$0.660895\pi$$
$$80$$ −150.093 −0.209762
$$81$$ 0 0
$$82$$ 787.990 1.06121
$$83$$ −196.997 −0.260521 −0.130261 0.991480i $$-0.541581\pi$$
−0.130261 + 0.991480i $$0.541581\pi$$
$$84$$ 0 0
$$85$$ 528.000 0.673760
$$86$$ −872.000 −1.09337
$$87$$ 0 0
$$88$$ 160.000 0.193819
$$89$$ 1500.93 1.78762 0.893812 0.448441i $$-0.148021\pi$$
0.893812 + 0.448441i $$0.148021\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −192.000 −0.217580
$$93$$ 0 0
$$94$$ 412.757 0.452900
$$95$$ −88.0000 −0.0950380
$$96$$ 0 0
$$97$$ −656.658 −0.687356 −0.343678 0.939088i $$-0.611673\pi$$
−0.343678 + 0.939088i $$0.611673\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ −148.000 −0.148000
$$101$$ −121.951 −0.120144 −0.0600721 0.998194i $$-0.519133\pi$$
−0.0600721 + 0.998194i $$0.519133\pi$$
$$102$$ 0 0
$$103$$ −1369.60 −1.31020 −0.655101 0.755541i $$-0.727374\pi$$
−0.655101 + 0.755541i $$0.727374\pi$$
$$104$$ −525.327 −0.495313
$$105$$ 0 0
$$106$$ 124.000 0.113622
$$107$$ 260.000 0.234908 0.117454 0.993078i $$-0.462527\pi$$
0.117454 + 0.993078i $$0.462527\pi$$
$$108$$ 0 0
$$109$$ 1882.00 1.65379 0.826894 0.562358i $$-0.190106\pi$$
0.826894 + 0.562358i $$0.190106\pi$$
$$110$$ −375.233 −0.325246
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 1286.00 1.07059 0.535295 0.844665i $$-0.320200\pi$$
0.535295 + 0.844665i $$0.320200\pi$$
$$114$$ 0 0
$$115$$ 450.280 0.365120
$$116$$ 664.000 0.531473
$$117$$ 0 0
$$118$$ −1332.08 −1.03922
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −931.000 −0.699474
$$122$$ −544.088 −0.403766
$$123$$ 0 0
$$124$$ −825.513 −0.597849
$$125$$ 1519.69 1.08741
$$126$$ 0 0
$$127$$ 2312.00 1.61541 0.807704 0.589588i $$-0.200710\pi$$
0.807704 + 0.589588i $$0.200710\pi$$
$$128$$ −128.000 −0.0883883
$$129$$ 0 0
$$130$$ 1232.00 0.831181
$$131$$ 253.282 0.168927 0.0844633 0.996427i $$-0.473082\pi$$
0.0844633 + 0.996427i $$0.473082\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ −1160.00 −0.747826
$$135$$ 0 0
$$136$$ 450.280 0.283906
$$137$$ 1114.00 0.694711 0.347356 0.937733i $$-0.387080\pi$$
0.347356 + 0.937733i $$0.387080\pi$$
$$138$$ 0 0
$$139$$ 1378.98 0.841466 0.420733 0.907185i $$-0.361773\pi$$
0.420733 + 0.907185i $$0.361773\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −1088.00 −0.642978
$$143$$ −1313.32 −0.768007
$$144$$ 0 0
$$145$$ −1557.22 −0.891862
$$146$$ 1200.75 0.680647
$$147$$ 0 0
$$148$$ −312.000 −0.173285
$$149$$ 946.000 0.520130 0.260065 0.965591i $$-0.416256\pi$$
0.260065 + 0.965591i $$0.416256\pi$$
$$150$$ 0 0
$$151$$ 832.000 0.448392 0.224196 0.974544i $$-0.428024\pi$$
0.224196 + 0.974544i $$0.428024\pi$$
$$152$$ −75.0467 −0.0400466
$$153$$ 0 0
$$154$$ 0 0
$$155$$ 1936.00 1.00325
$$156$$ 0 0
$$157$$ 2879.92 1.46396 0.731982 0.681324i $$-0.238596\pi$$
0.731982 + 0.681324i $$0.238596\pi$$
$$158$$ 1360.00 0.684783
$$159$$ 0 0
$$160$$ 300.187 0.148324
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 636.000 0.305616 0.152808 0.988256i $$-0.451168\pi$$
0.152808 + 0.988256i $$0.451168\pi$$
$$164$$ −1575.98 −0.750386
$$165$$ 0 0
$$166$$ 393.995 0.184216
$$167$$ 656.658 0.304274 0.152137 0.988359i $$-0.451385\pi$$
0.152137 + 0.988359i $$0.451385\pi$$
$$168$$ 0 0
$$169$$ 2115.00 0.962676
$$170$$ −1056.00 −0.476421
$$171$$ 0 0
$$172$$ 1744.00 0.773132
$$173$$ −666.039 −0.292705 −0.146353 0.989232i $$-0.546753\pi$$
−0.146353 + 0.989232i $$0.546753\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −320.000 −0.137051
$$177$$ 0 0
$$178$$ −3001.87 −1.26404
$$179$$ 3228.00 1.34789 0.673944 0.738782i $$-0.264599\pi$$
0.673944 + 0.738782i $$0.264599\pi$$
$$180$$ 0 0
$$181$$ −2823.63 −1.15955 −0.579776 0.814776i $$-0.696860\pi$$
−0.579776 + 0.814776i $$0.696860\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ 384.000 0.153852
$$185$$ 731.705 0.290789
$$186$$ 0 0
$$187$$ 1125.70 0.440210
$$188$$ −825.513 −0.320249
$$189$$ 0 0
$$190$$ 176.000 0.0672020
$$191$$ 2136.00 0.809191 0.404596 0.914496i $$-0.367412\pi$$
0.404596 + 0.914496i $$0.367412\pi$$
$$192$$ 0 0
$$193$$ 1658.00 0.618370 0.309185 0.951002i $$-0.399944\pi$$
0.309185 + 0.951002i $$0.399944\pi$$
$$194$$ 1313.32 0.486034
$$195$$ 0 0
$$196$$ 0 0
$$197$$ 978.000 0.353704 0.176852 0.984237i $$-0.443409\pi$$
0.176852 + 0.984237i $$0.443409\pi$$
$$198$$ 0 0
$$199$$ −4934.32 −1.75771 −0.878855 0.477088i $$-0.841692\pi$$
−0.878855 + 0.477088i $$0.841692\pi$$
$$200$$ 296.000 0.104652
$$201$$ 0 0
$$202$$ 243.902 0.0849547
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 3696.00 1.25922
$$206$$ 2739.20 0.926453
$$207$$ 0 0
$$208$$ 1050.65 0.350239
$$209$$ −187.617 −0.0620943
$$210$$ 0 0
$$211$$ 1556.00 0.507675 0.253838 0.967247i $$-0.418307\pi$$
0.253838 + 0.967247i $$0.418307\pi$$
$$212$$ −248.000 −0.0803430
$$213$$ 0 0
$$214$$ −520.000 −0.166105
$$215$$ −4090.04 −1.29739
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −3764.00 −1.16940
$$219$$ 0 0
$$220$$ 750.467 0.229984
$$221$$ −3696.00 −1.12498
$$222$$ 0 0
$$223$$ 2889.30 0.867630 0.433815 0.901002i $$-0.357167\pi$$
0.433815 + 0.901002i $$0.357167\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −2572.00 −0.757022
$$227$$ 1979.36 0.578742 0.289371 0.957217i $$-0.406554\pi$$
0.289371 + 0.957217i $$0.406554\pi$$
$$228$$ 0 0
$$229$$ 2767.35 0.798565 0.399282 0.916828i $$-0.369259\pi$$
0.399282 + 0.916828i $$0.369259\pi$$
$$230$$ −900.560 −0.258179
$$231$$ 0 0
$$232$$ −1328.00 −0.375808
$$233$$ 6490.00 1.82478 0.912391 0.409321i $$-0.134234\pi$$
0.912391 + 0.409321i $$0.134234\pi$$
$$234$$ 0 0
$$235$$ 1936.00 0.537407
$$236$$ 2664.16 0.734838
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 4296.00 1.16270 0.581350 0.813654i $$-0.302525\pi$$
0.581350 + 0.813654i $$0.302525\pi$$
$$240$$ 0 0
$$241$$ 4521.56 1.20854 0.604272 0.796778i $$-0.293464\pi$$
0.604272 + 0.796778i $$0.293464\pi$$
$$242$$ 1862.00 0.494603
$$243$$ 0 0
$$244$$ 1088.18 0.285506
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 616.000 0.158685
$$248$$ 1651.03 0.422743
$$249$$ 0 0
$$250$$ −3039.39 −0.768911
$$251$$ −5581.59 −1.40361 −0.701807 0.712367i $$-0.747623\pi$$
−0.701807 + 0.712367i $$0.747623\pi$$
$$252$$ 0 0
$$253$$ 960.000 0.238556
$$254$$ −4624.00 −1.14227
$$255$$ 0 0
$$256$$ 256.000 0.0625000
$$257$$ 1500.93 0.364302 0.182151 0.983271i $$-0.441694\pi$$
0.182151 + 0.983271i $$0.441694\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ −2464.00 −0.587734
$$261$$ 0 0
$$262$$ −506.565 −0.119449
$$263$$ 400.000 0.0937835 0.0468917 0.998900i $$-0.485068\pi$$
0.0468917 + 0.998900i $$0.485068\pi$$
$$264$$ 0 0
$$265$$ 581.612 0.134823
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 2320.00 0.528793
$$269$$ 272.044 0.0616610 0.0308305 0.999525i $$-0.490185\pi$$
0.0308305 + 0.999525i $$0.490185\pi$$
$$270$$ 0 0
$$271$$ 6904.29 1.54762 0.773812 0.633416i $$-0.218348\pi$$
0.773812 + 0.633416i $$0.218348\pi$$
$$272$$ −900.560 −0.200752
$$273$$ 0 0
$$274$$ −2228.00 −0.491235
$$275$$ 740.000 0.162268
$$276$$ 0 0
$$277$$ −6770.00 −1.46848 −0.734242 0.678888i $$-0.762462\pi$$
−0.734242 + 0.678888i $$0.762462\pi$$
$$278$$ −2757.96 −0.595006
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −1878.00 −0.398691 −0.199345 0.979929i $$-0.563882\pi$$
−0.199345 + 0.979929i $$0.563882\pi$$
$$282$$ 0 0
$$283$$ 384.614 0.0807878 0.0403939 0.999184i $$-0.487139\pi$$
0.0403939 + 0.999184i $$0.487139\pi$$
$$284$$ 2176.00 0.454654
$$285$$ 0 0
$$286$$ 2626.63 0.543063
$$287$$ 0 0
$$288$$ 0 0
$$289$$ −1745.00 −0.355180
$$290$$ 3114.44 0.630641
$$291$$ 0 0
$$292$$ −2401.49 −0.481290
$$293$$ 3742.95 0.746299 0.373149 0.927771i $$-0.378278\pi$$
0.373149 + 0.927771i $$0.378278\pi$$
$$294$$ 0 0
$$295$$ −6248.00 −1.23313
$$296$$ 624.000 0.122531
$$297$$ 0 0
$$298$$ −1892.00 −0.367787
$$299$$ −3151.96 −0.609641
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −1664.00 −0.317061
$$303$$ 0 0
$$304$$ 150.093 0.0283172
$$305$$ −2552.00 −0.479105
$$306$$ 0 0
$$307$$ 722.324 0.134284 0.0671420 0.997743i $$-0.478612\pi$$
0.0671420 + 0.997743i $$0.478612\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −3872.00 −0.709403
$$311$$ −7279.53 −1.32728 −0.663640 0.748052i $$-0.730989\pi$$
−0.663640 + 0.748052i $$0.730989\pi$$
$$312$$ 0 0
$$313$$ 1519.69 0.274435 0.137218 0.990541i $$-0.456184\pi$$
0.137218 + 0.990541i $$0.456184\pi$$
$$314$$ −5759.83 −1.03518
$$315$$ 0 0
$$316$$ −2720.00 −0.484215
$$317$$ −2358.00 −0.417787 −0.208893 0.977938i $$-0.566986\pi$$
−0.208893 + 0.977938i $$0.566986\pi$$
$$318$$ 0 0
$$319$$ −3320.00 −0.582709
$$320$$ −600.373 −0.104881
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −528.000 −0.0909557
$$324$$ 0 0
$$325$$ −2429.64 −0.414683
$$326$$ −1272.00 −0.216103
$$327$$ 0 0
$$328$$ 3151.96 0.530603
$$329$$ 0 0
$$330$$ 0 0
$$331$$ 2372.00 0.393888 0.196944 0.980415i $$-0.436898\pi$$
0.196944 + 0.980415i $$0.436898\pi$$
$$332$$ −787.990 −0.130261
$$333$$ 0 0
$$334$$ −1313.32 −0.215154
$$335$$ −5440.88 −0.887365
$$336$$ 0 0
$$337$$ −250.000 −0.0404106 −0.0202053 0.999796i $$-0.506432\pi$$
−0.0202053 + 0.999796i $$0.506432\pi$$
$$338$$ −4230.00 −0.680715
$$339$$ 0 0
$$340$$ 2112.00 0.336880
$$341$$ 4127.57 0.655485
$$342$$ 0 0
$$343$$ 0 0
$$344$$ −3488.00 −0.546687
$$345$$ 0 0
$$346$$ 1332.08 0.206974
$$347$$ −9540.00 −1.47589 −0.737945 0.674861i $$-0.764204\pi$$
−0.737945 + 0.674861i $$0.764204\pi$$
$$348$$ 0 0
$$349$$ −5712.93 −0.876235 −0.438117 0.898918i $$-0.644355\pi$$
−0.438117 + 0.898918i $$0.644355\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 640.000 0.0969094
$$353$$ −4390.23 −0.661950 −0.330975 0.943640i $$-0.607378\pi$$
−0.330975 + 0.943640i $$0.607378\pi$$
$$354$$ 0 0
$$355$$ −5103.17 −0.762953
$$356$$ 6003.73 0.893812
$$357$$ 0 0
$$358$$ −6456.00 −0.953101
$$359$$ −1840.00 −0.270506 −0.135253 0.990811i $$-0.543185\pi$$
−0.135253 + 0.990811i $$0.543185\pi$$
$$360$$ 0 0
$$361$$ −6771.00 −0.987170
$$362$$ 5647.26 0.819927
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 5632.00 0.807650
$$366$$ 0 0
$$367$$ −2964.34 −0.421628 −0.210814 0.977526i $$-0.567611\pi$$
−0.210814 + 0.977526i $$0.567611\pi$$
$$368$$ −768.000 −0.108790
$$369$$ 0 0
$$370$$ −1463.41 −0.205619
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 3982.00 0.552762 0.276381 0.961048i $$-0.410865\pi$$
0.276381 + 0.961048i $$0.410865\pi$$
$$374$$ −2251.40 −0.311276
$$375$$ 0 0
$$376$$ 1651.03 0.226450
$$377$$ 10900.5 1.48914
$$378$$ 0 0
$$379$$ 2676.00 0.362683 0.181342 0.983420i $$-0.441956\pi$$
0.181342 + 0.983420i $$0.441956\pi$$
$$380$$ −352.000 −0.0475190
$$381$$ 0 0
$$382$$ −4272.00 −0.572185
$$383$$ −7035.62 −0.938652 −0.469326 0.883025i $$-0.655503\pi$$
−0.469326 + 0.883025i $$0.655503\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −3316.00 −0.437254
$$387$$ 0 0
$$388$$ −2626.63 −0.343678
$$389$$ −8658.00 −1.12848 −0.564239 0.825611i $$-0.690830\pi$$
−0.564239 + 0.825611i $$0.690830\pi$$
$$390$$ 0 0
$$391$$ 2701.68 0.349437
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −1956.00 −0.250106
$$395$$ 6378.97 0.812558
$$396$$ 0 0
$$397$$ −9052.50 −1.14441 −0.572207 0.820109i $$-0.693912\pi$$
−0.572207 + 0.820109i $$0.693912\pi$$
$$398$$ 9868.63 1.24289
$$399$$ 0 0
$$400$$ −592.000 −0.0740000
$$401$$ 5706.00 0.710584 0.355292 0.934755i $$-0.384381\pi$$
0.355292 + 0.934755i $$0.384381\pi$$
$$402$$ 0 0
$$403$$ −13552.0 −1.67512
$$404$$ −487.803 −0.0600721
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 1560.00 0.189991
$$408$$ 0 0
$$409$$ 2420.25 0.292601 0.146301 0.989240i $$-0.453263\pi$$
0.146301 + 0.989240i $$0.453263\pi$$
$$410$$ −7392.00 −0.890402
$$411$$ 0 0
$$412$$ −5478.41 −0.655101
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 1848.00 0.218590
$$416$$ −2101.31 −0.247656
$$417$$ 0 0
$$418$$ 375.233 0.0439073
$$419$$ 1510.31 0.176095 0.0880473 0.996116i $$-0.471937\pi$$
0.0880473 + 0.996116i $$0.471937\pi$$
$$420$$ 0 0
$$421$$ −16770.0 −1.94138 −0.970689 0.240341i $$-0.922741\pi$$
−0.970689 + 0.240341i $$0.922741\pi$$
$$422$$ −3112.00 −0.358981
$$423$$ 0 0
$$424$$ 496.000 0.0568111
$$425$$ 2082.54 0.237690
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 1040.00 0.117454
$$429$$ 0 0
$$430$$ 8180.09 0.917392
$$431$$ −1336.00 −0.149311 −0.0746553 0.997209i $$-0.523786\pi$$
−0.0746553 + 0.997209i $$0.523786\pi$$
$$432$$ 0 0
$$433$$ −11163.2 −1.23896 −0.619479 0.785013i $$-0.712656\pi$$
−0.619479 + 0.785013i $$0.712656\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 7528.00 0.826894
$$437$$ −450.280 −0.0492902
$$438$$ 0 0
$$439$$ −3602.24 −0.391630 −0.195815 0.980641i $$-0.562735\pi$$
−0.195815 + 0.980641i $$0.562735\pi$$
$$440$$ −1500.93 −0.162623
$$441$$ 0 0
$$442$$ 7392.00 0.795479
$$443$$ −6348.00 −0.680818 −0.340409 0.940277i $$-0.610566\pi$$
−0.340409 + 0.940277i $$0.610566\pi$$
$$444$$ 0 0
$$445$$ −14080.0 −1.49990
$$446$$ −5778.59 −0.613507
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −7170.00 −0.753615 −0.376808 0.926292i $$-0.622978\pi$$
−0.376808 + 0.926292i $$0.622978\pi$$
$$450$$ 0 0
$$451$$ 7879.90 0.822727
$$452$$ 5144.00 0.535295
$$453$$ 0 0
$$454$$ −3958.71 −0.409232
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 6866.00 0.702796 0.351398 0.936226i $$-0.385706\pi$$
0.351398 + 0.936226i $$0.385706\pi$$
$$458$$ −5534.69 −0.564671
$$459$$ 0 0
$$460$$ 1801.12 0.182560
$$461$$ −1378.98 −0.139318 −0.0696590 0.997571i $$-0.522191\pi$$
−0.0696590 + 0.997571i $$0.522191\pi$$
$$462$$ 0 0
$$463$$ 2648.00 0.265795 0.132897 0.991130i $$-0.457572\pi$$
0.132897 + 0.991130i $$0.457572\pi$$
$$464$$ 2656.00 0.265736
$$465$$ 0 0
$$466$$ −12980.0 −1.29032
$$467$$ 12335.8 1.22234 0.611170 0.791500i $$-0.290699\pi$$
0.611170 + 0.791500i $$0.290699\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ −3872.00 −0.380004
$$471$$ 0 0
$$472$$ −5328.31 −0.519609
$$473$$ −8720.00 −0.847666
$$474$$ 0 0
$$475$$ −347.091 −0.0335276
$$476$$ 0 0
$$477$$ 0 0
$$478$$ −8592.00 −0.822153
$$479$$ 13339.5 1.27244 0.636221 0.771507i $$-0.280497\pi$$
0.636221 + 0.771507i $$0.280497\pi$$
$$480$$ 0 0
$$481$$ −5121.93 −0.485530
$$482$$ −9043.12 −0.854570
$$483$$ 0 0
$$484$$ −3724.00 −0.349737
$$485$$ 6160.00 0.576724
$$486$$ 0 0
$$487$$ 13936.0 1.29672 0.648358 0.761336i $$-0.275456\pi$$
0.648358 + 0.761336i $$0.275456\pi$$
$$488$$ −2176.35 −0.201883
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 12276.0 1.12833 0.564163 0.825663i $$-0.309199\pi$$
0.564163 + 0.825663i $$0.309199\pi$$
$$492$$ 0 0
$$493$$ −9343.31 −0.853553
$$494$$ −1232.00 −0.112207
$$495$$ 0 0
$$496$$ −3302.05 −0.298924
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −2220.00 −0.199160 −0.0995800 0.995030i $$-0.531750\pi$$
−0.0995800 + 0.995030i $$0.531750\pi$$
$$500$$ 6078.78 0.543703
$$501$$ 0 0
$$502$$ 11163.2 0.992505
$$503$$ 11294.5 1.00119 0.500594 0.865682i $$-0.333115\pi$$
0.500594 + 0.865682i $$0.333115\pi$$
$$504$$ 0 0
$$505$$ 1144.00 0.100807
$$506$$ −1920.00 −0.168685
$$507$$ 0 0
$$508$$ 9248.00 0.807704
$$509$$ 15881.7 1.38300 0.691499 0.722377i $$-0.256951\pi$$
0.691499 + 0.722377i $$0.256951\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −512.000 −0.0441942
$$513$$ 0 0
$$514$$ −3001.87 −0.257600
$$515$$ 12848.0 1.09932
$$516$$ 0 0
$$517$$ 4127.57 0.351122
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 4928.00 0.415591
$$521$$ −11613.5 −0.976575 −0.488287 0.872683i $$-0.662378\pi$$
−0.488287 + 0.872683i $$0.662378\pi$$
$$522$$ 0 0
$$523$$ 12617.2 1.05490 0.527450 0.849586i $$-0.323148\pi$$
0.527450 + 0.849586i $$0.323148\pi$$
$$524$$ 1013.13 0.0844633
$$525$$ 0 0
$$526$$ −800.000 −0.0663149
$$527$$ 11616.0 0.960154
$$528$$ 0 0
$$529$$ −9863.00 −0.810635
$$530$$ −1163.22 −0.0953343
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −25872.0 −2.10252
$$534$$ 0 0
$$535$$ −2439.02 −0.197099
$$536$$ −4640.00 −0.373913
$$537$$ 0 0
$$538$$ −544.088 −0.0436009
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 1798.00 0.142887 0.0714437 0.997445i $$-0.477239\pi$$
0.0714437 + 0.997445i $$0.477239\pi$$
$$542$$ −13808.6 −1.09433
$$543$$ 0 0
$$544$$ 1801.12 0.141953
$$545$$ −17654.7 −1.38761
$$546$$ 0 0
$$547$$ 1276.00 0.0997401 0.0498700 0.998756i $$-0.484119\pi$$
0.0498700 + 0.998756i $$0.484119\pi$$
$$548$$ 4456.00 0.347356
$$549$$ 0 0
$$550$$ −1480.00 −0.114741
$$551$$ 1557.22 0.120399
$$552$$ 0 0
$$553$$ 0 0
$$554$$ 13540.0 1.03837
$$555$$ 0 0
$$556$$ 5515.93 0.420733
$$557$$ −2694.00 −0.204934 −0.102467 0.994736i $$-0.532674\pi$$
−0.102467 + 0.994736i $$0.532674\pi$$
$$558$$ 0 0
$$559$$ 28630.3 2.16625
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 3756.00 0.281917
$$563$$ 15769.2 1.18045 0.590223 0.807240i $$-0.299040\pi$$
0.590223 + 0.807240i $$0.299040\pi$$
$$564$$ 0 0
$$565$$ −12063.7 −0.898276
$$566$$ −769.228 −0.0571256
$$567$$ 0 0
$$568$$ −4352.00 −0.321489
$$569$$ −12606.0 −0.928772 −0.464386 0.885633i $$-0.653725\pi$$
−0.464386 + 0.885633i $$0.653725\pi$$
$$570$$ 0 0
$$571$$ 6852.00 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ −5253.27 −0.384004
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 1776.00 0.128808
$$576$$ 0 0
$$577$$ 14371.4 1.03690 0.518449 0.855108i $$-0.326509\pi$$
0.518449 + 0.855108i $$0.326509\pi$$
$$578$$ 3490.00 0.251150
$$579$$ 0 0
$$580$$ −6228.87 −0.445931
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 1240.00 0.0880884
$$584$$ 4802.99 0.340324
$$585$$ 0 0
$$586$$ −7485.90 −0.527713
$$587$$ −18977.4 −1.33438 −0.667191 0.744887i $$-0.732503\pi$$
−0.667191 + 0.744887i $$0.732503\pi$$
$$588$$ 0 0
$$589$$ −1936.00 −0.135435
$$590$$ 12496.0 0.871953
$$591$$ 0 0
$$592$$ −1248.00 −0.0866427
$$593$$ −8217.61 −0.569067 −0.284534 0.958666i $$-0.591839\pi$$
−0.284534 + 0.958666i $$0.591839\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 3784.00 0.260065
$$597$$ 0 0
$$598$$ 6303.92 0.431081
$$599$$ 19104.0 1.30312 0.651559 0.758598i $$-0.274115\pi$$
0.651559 + 0.758598i $$0.274115\pi$$
$$600$$ 0 0
$$601$$ 21538.4 1.46185 0.730923 0.682460i $$-0.239090\pi$$
0.730923 + 0.682460i $$0.239090\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ 3328.00 0.224196
$$605$$ 8733.55 0.586892
$$606$$ 0 0
$$607$$ 13733.5 0.918331 0.459166 0.888351i $$-0.348148\pi$$
0.459166 + 0.888351i $$0.348148\pi$$
$$608$$ −300.187 −0.0200233
$$609$$ 0 0
$$610$$ 5104.00 0.338779
$$611$$ −13552.0 −0.897308
$$612$$ 0 0
$$613$$ 28034.0 1.84712 0.923558 0.383458i $$-0.125267\pi$$
0.923558 + 0.383458i $$0.125267\pi$$
$$614$$ −1444.65 −0.0949532
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 8258.00 0.538824 0.269412 0.963025i $$-0.413171\pi$$
0.269412 + 0.963025i $$0.413171\pi$$
$$618$$ 0 0
$$619$$ −5131.31 −0.333191 −0.166595 0.986025i $$-0.553277\pi$$
−0.166595 + 0.986025i $$0.553277\pi$$
$$620$$ 7744.00 0.501623
$$621$$ 0 0
$$622$$ 14559.1 0.938529
$$623$$ 0 0
$$624$$ 0 0
$$625$$ −9631.00 −0.616384
$$626$$ −3039.39 −0.194055
$$627$$ 0 0
$$628$$ 11519.7 0.731982
$$629$$ 4390.23 0.278299
$$630$$ 0 0
$$631$$ 912.000 0.0575375 0.0287687 0.999586i $$-0.490841\pi$$
0.0287687 + 0.999586i $$0.490841\pi$$
$$632$$ 5440.00 0.342392
$$633$$ 0 0
$$634$$ 4716.00 0.295420
$$635$$ −21688.5 −1.35540
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 6640.00 0.412038
$$639$$ 0 0
$$640$$ 1200.75 0.0741620
$$641$$ 890.000 0.0548407 0.0274203 0.999624i $$-0.491271\pi$$
0.0274203 + 0.999624i $$0.491271\pi$$
$$642$$ 0 0
$$643$$ −29352.6 −1.80024 −0.900120 0.435642i $$-0.856521\pi$$
−0.900120 + 0.435642i $$0.856521\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 1056.00 0.0643154
$$647$$ −11876.1 −0.721637 −0.360818 0.932636i $$-0.617503\pi$$
−0.360818 + 0.932636i $$0.617503\pi$$
$$648$$ 0 0
$$649$$ −13320.8 −0.805680
$$650$$ 4859.27 0.293225
$$651$$ 0 0
$$652$$ 2544.00 0.152808
$$653$$ 21526.0 1.29001 0.645006 0.764178i $$-0.276855\pi$$
0.645006 + 0.764178i $$0.276855\pi$$
$$654$$ 0 0
$$655$$ −2376.00 −0.141737
$$656$$ −6303.92 −0.375193
$$657$$ 0 0
$$658$$ 0 0
$$659$$ −23452.0 −1.38628 −0.693141 0.720802i $$-0.743774\pi$$
−0.693141 + 0.720802i $$0.743774\pi$$
$$660$$ 0 0
$$661$$ −26669.7 −1.56934 −0.784668 0.619916i $$-0.787167\pi$$
−0.784668 + 0.619916i $$0.787167\pi$$
$$662$$ −4744.00 −0.278521
$$663$$ 0 0
$$664$$ 1575.98 0.0921082
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −7968.00 −0.462552
$$668$$ 2626.63 0.152137
$$669$$ 0 0
$$670$$ 10881.8 0.627462
$$671$$ −5440.88 −0.313030
$$672$$ 0 0
$$673$$ −13858.0 −0.793739 −0.396870 0.917875i $$-0.629904\pi$$
−0.396870 + 0.917875i $$0.629904\pi$$
$$674$$ 500.000 0.0285746
$$675$$ 0 0
$$676$$ 8460.00 0.481338
$$677$$ −32448.3 −1.84208 −0.921041 0.389466i $$-0.872660\pi$$
−0.921041 + 0.389466i $$0.872660\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −4224.00 −0.238210
$$681$$ 0 0
$$682$$ −8255.13 −0.463498
$$683$$ 27812.0 1.55812 0.779060 0.626949i $$-0.215696\pi$$
0.779060 + 0.626949i $$0.215696\pi$$
$$684$$ 0 0
$$685$$ −10450.2 −0.582895
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 6976.00 0.386566
$$689$$ −4071.28 −0.225114
$$690$$ 0 0
$$691$$ −1303.94 −0.0717859 −0.0358929 0.999356i $$-0.511428\pi$$
−0.0358929 + 0.999356i $$0.511428\pi$$
$$692$$ −2664.16 −0.146353
$$693$$ 0 0
$$694$$ 19080.0 1.04361
$$695$$ −12936.0 −0.706029
$$696$$ 0 0
$$697$$ 22176.0 1.20513
$$698$$ 11425.9 0.619592
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −22906.0 −1.23416 −0.617081 0.786900i $$-0.711685\pi$$
−0.617081 + 0.786900i $$0.711685\pi$$
$$702$$ 0 0
$$703$$ −731.705 −0.0392557
$$704$$ −1280.00 −0.0685253
$$705$$ 0 0
$$706$$ 8780.46 0.468069
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −15086.0 −0.799107 −0.399553 0.916710i $$-0.630835\pi$$
−0.399553 + 0.916710i $$0.630835\pi$$
$$710$$ 10206.3 0.539489
$$711$$ 0 0
$$712$$ −12007.5 −0.632021
$$713$$ 9906.16 0.520321
$$714$$ 0 0
$$715$$ 12320.0 0.644394
$$716$$ 12912.0 0.673944
$$717$$ 0 0
$$718$$ 3680.00 0.191276
$$719$$ 20544.0 1.06559 0.532797 0.846243i $$-0.321141\pi$$
0.532797 + 0.846243i $$0.321141\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 13542.0 0.698035
$$723$$ 0 0
$$724$$ −11294.5 −0.579776
$$725$$ −6142.00 −0.314632
$$726$$ 0 0
$$727$$ −7223.24 −0.368494 −0.184247 0.982880i $$-0.558985\pi$$
−0.184247 + 0.982880i $$0.558985\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ −11264.0 −0.571095
$$731$$ −24540.3 −1.24166
$$732$$ 0 0
$$733$$ 29427.7 1.48286 0.741430 0.671031i $$-0.234148\pi$$
0.741430 + 0.671031i $$0.234148\pi$$
$$734$$ 5928.69 0.298136
$$735$$ 0 0
$$736$$ 1536.00 0.0769262
$$737$$ −11600.0 −0.579771
$$738$$ 0 0
$$739$$ 32668.0 1.62613 0.813066 0.582171i $$-0.197797\pi$$
0.813066 + 0.582171i $$0.197797\pi$$
$$740$$ 2926.82 0.145395
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 37056.0 1.82968 0.914840 0.403816i $$-0.132316\pi$$
0.914840 + 0.403816i $$0.132316\pi$$
$$744$$ 0 0
$$745$$ −8874.27 −0.436413
$$746$$ −7964.00 −0.390862
$$747$$ 0 0
$$748$$ 4502.80 0.220105
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −19608.0 −0.952738 −0.476369 0.879246i $$-0.658047\pi$$
−0.476369 + 0.879246i $$0.658047\pi$$
$$752$$ −3302.05 −0.160124
$$753$$ 0 0
$$754$$ −21801.1 −1.05298
$$755$$ −7804.85 −0.376222
$$756$$ 0 0
$$757$$ 19378.0 0.930390 0.465195 0.885208i $$-0.345984\pi$$
0.465195 + 0.885208i $$0.345984\pi$$
$$758$$ −5352.00 −0.256456
$$759$$ 0 0
$$760$$ 704.000 0.0336010
$$761$$ 13977.4 0.665810 0.332905 0.942960i $$-0.391971\pi$$
0.332905 + 0.942960i $$0.391971\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 8544.00 0.404596
$$765$$ 0 0
$$766$$ 14071.2 0.663727
$$767$$ 43736.0 2.05895
$$768$$ 0 0
$$769$$ 8536.56 0.400307 0.200154 0.979765i $$-0.435856\pi$$
0.200154 + 0.979765i $$0.435856\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 6632.00 0.309185
$$773$$ 29296.3 1.36315 0.681576 0.731748i $$-0.261295\pi$$
0.681576 + 0.731748i $$0.261295\pi$$
$$774$$ 0 0
$$775$$ 7636.00 0.353927
$$776$$ 5253.27 0.243017
$$777$$ 0 0
$$778$$ 17316.0 0.797955
$$779$$ −3696.00 −0.169991
$$780$$ 0 0
$$781$$ −10880.0 −0.498485
$$782$$ −5403.36 −0.247089
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −27016.0 −1.22833
$$786$$ 0 0
$$787$$ 13780.4 0.624167 0.312084 0.950055i $$-0.398973\pi$$
0.312084 + 0.950055i $$0.398973\pi$$
$$788$$ 3912.00 0.176852
$$789$$ 0 0
$$790$$ −12757.9 −0.574566
$$791$$ 0 0
$$792$$ 0 0
$$793$$ 17864.0 0.799961
$$794$$ 18105.0 0.809222
$$795$$ 0 0
$$796$$ −19737.3 −0.878855
$$797$$ 34868.6 1.54970 0.774848 0.632148i $$-0.217826\pi$$
0.774848 + 0.632148i $$0.217826\pi$$
$$798$$ 0 0
$$799$$ 11616.0 0.514324
$$800$$ 1184.00 0.0523259
$$801$$ 0 0
$$802$$ −11412.0 −0.502459
$$803$$ 12007.5 0.527689
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 27104.0 1.18449
$$807$$ 0 0
$$808$$ 975.606 0.0424774
$$809$$ −14034.0 −0.609900 −0.304950 0.952368i $$-0.598640\pi$$
−0.304950 + 0.952368i $$0.598640\pi$$
$$810$$ 0 0
$$811$$ −6632.25 −0.287164 −0.143582 0.989638i $$-0.545862\pi$$
−0.143582 + 0.989638i $$0.545862\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ −3120.00 −0.134344
$$815$$ −5966.21 −0.256426
$$816$$ 0 0
$$817$$ 4090.04 0.175144
$$818$$ −4840.51 −0.206900
$$819$$ 0 0
$$820$$ 14784.0 0.629609
$$821$$ −28622.0 −1.21670 −0.608352 0.793667i $$-0.708169\pi$$
−0.608352 + 0.793667i $$0.708169\pi$$
$$822$$ 0 0
$$823$$ 24688.0 1.04565 0.522825 0.852440i $$-0.324878\pi$$
0.522825 + 0.852440i $$0.324878\pi$$
$$824$$ 10956.8 0.463226
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 30756.0 1.29322 0.646609 0.762822i $$-0.276187\pi$$
0.646609 + 0.762822i $$0.276187\pi$$
$$828$$ 0 0
$$829$$ 23236.3 0.973499 0.486750 0.873542i $$-0.338182\pi$$
0.486750 + 0.873542i $$0.338182\pi$$
$$830$$ −3696.00 −0.154566
$$831$$ 0 0
$$832$$ 4202.61 0.175119
$$833$$ 0 0
$$834$$ 0 0
$$835$$ −6160.00 −0.255300
$$836$$ −750.467 −0.0310472
$$837$$ 0 0
$$838$$ −3020.63 −0.124518
$$839$$ 24033.7 0.988957 0.494479 0.869190i $$-0.335359\pi$$
0.494479 + 0.869190i $$0.335359\pi$$
$$840$$ 0 0
$$841$$ 3167.00 0.129854
$$842$$ 33540.0 1.37276
$$843$$ 0 0
$$844$$ 6224.00 0.253838
$$845$$ −19840.5 −0.807731
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −992.000 −0.0401715
$$849$$ 0 0
$$850$$ −4165.09 −0.168072
$$851$$ 3744.00 0.150814
$$852$$ 0 0
$$853$$ −23574.0 −0.946260 −0.473130 0.880993i $$-0.656876\pi$$
−0.473130 + 0.880993i $$0.656876\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −2080.00 −0.0830525
$$857$$ −24484.0 −0.975912 −0.487956 0.872868i $$-0.662257\pi$$
−0.487956 + 0.872868i $$0.662257\pi$$
$$858$$ 0 0
$$859$$ −32954.9 −1.30897 −0.654485 0.756075i $$-0.727115\pi$$
−0.654485 + 0.756075i $$0.727115\pi$$
$$860$$ −16360.2 −0.648694
$$861$$ 0 0
$$862$$ 2672.00 0.105579
$$863$$ −40872.0 −1.61217 −0.806083 0.591803i $$-0.798416\pi$$
−0.806083 + 0.591803i $$0.798416\pi$$
$$864$$ 0 0
$$865$$ 6248.00 0.245593
$$866$$ 22326.4 0.876075
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 13600.0 0.530896
$$870$$ 0 0
$$871$$ 38086.2 1.48163
$$872$$ −15056.0 −0.584702
$$873$$ 0 0
$$874$$ 900.560 0.0348534
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −12006.0 −0.462273 −0.231137 0.972921i $$-0.574244\pi$$
−0.231137 + 0.972921i $$0.574244\pi$$
$$878$$ 7204.48 0.276924
$$879$$ 0 0
$$880$$ 3001.87 0.114992
$$881$$ −35722.2 −1.36607 −0.683037 0.730383i $$-0.739341\pi$$
−0.683037 + 0.730383i $$0.739341\pi$$
$$882$$ 0 0
$$883$$ 19588.0 0.746533 0.373267 0.927724i $$-0.378238\pi$$
0.373267 + 0.927724i $$0.378238\pi$$
$$884$$ −14784.0 −0.562488
$$885$$ 0 0
$$886$$ 12696.0 0.481411
$$887$$ 40243.8 1.52340 0.761699 0.647931i $$-0.224366\pi$$
0.761699 + 0.647931i $$0.224366\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 28160.0 1.06059
$$891$$ 0 0
$$892$$ 11557.2 0.433815
$$893$$ −1936.00 −0.0725485
$$894$$ 0 0
$$895$$ −30281.3 −1.13094
$$896$$ 0 0
$$897$$ 0 0
$$898$$ 14340.0 0.532886
$$899$$ −34258.8 −1.27096
$$900$$ 0 0
$$901$$ 3489.67 0.129032
$$902$$ −15759.8 −0.581756
$$903$$ 0 0
$$904$$ −10288.0 −0.378511
$$905$$ 26488.0 0.972918
$$906$$ 0 0
$$907$$ 15868.0 0.580913 0.290457 0.956888i $$-0.406193\pi$$
0.290457 + 0.956888i $$0.406193\pi$$
$$908$$ 7917.42 0.289371
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −39832.0 −1.44862 −0.724310 0.689474i $$-0.757842\pi$$
−0.724310 + 0.689474i $$0.757842\pi$$
$$912$$ 0 0
$$913$$ 3939.95 0.142818
$$914$$ −13732.0 −0.496952
$$915$$ 0 0
$$916$$ 11069.4 0.399282
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −30528.0 −1.09578 −0.547892 0.836549i $$-0.684570\pi$$
−0.547892 + 0.836549i $$0.684570\pi$$
$$920$$ −3602.24 −0.129089
$$921$$ 0 0
$$922$$ 2757.96 0.0985127
$$923$$ 35722.2 1.27390
$$924$$ 0 0
$$925$$ 2886.00 0.102585
$$926$$ −5296.00 −0.187945
$$927$$ 0 0
$$928$$ −5312.00 −0.187904
$$929$$ 16604.1 0.586396 0.293198 0.956052i $$-0.405280\pi$$
0.293198 + 0.956052i $$0.405280\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 25960.0 0.912391
$$933$$ 0 0
$$934$$ −24671.6 −0.864324
$$935$$ −10560.0 −0.369357
$$936$$ 0 0
$$937$$ 29943.6 1.04399 0.521993 0.852950i $$-0.325189\pi$$
0.521993 + 0.852950i $$0.325189\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 7744.00 0.268704
$$941$$ −5375.22 −0.186214 −0.0931068 0.995656i $$-0.529680\pi$$
−0.0931068 + 0.995656i $$0.529680\pi$$
$$942$$ 0 0
$$943$$ 18911.8 0.653077
$$944$$ 10656.6 0.367419
$$945$$ 0 0
$$946$$ 17440.0 0.599390
$$947$$ −45212.0 −1.55142 −0.775709 0.631091i $$-0.782607\pi$$
−0.775709 + 0.631091i $$0.782607\pi$$
$$948$$ 0 0
$$949$$ −39424.0 −1.34853
$$950$$ 694.182 0.0237076
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −34218.0 −1.16310 −0.581548 0.813512i $$-0.697553\pi$$
−0.581548 + 0.813512i $$0.697553\pi$$
$$954$$ 0 0
$$955$$ −20037.5 −0.678950
$$956$$ 17184.0 0.581350
$$957$$ 0 0
$$958$$ −26679.1 −0.899752
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 12801.0 0.429694
$$962$$ 10243.9 0.343322
$$963$$ 0 0
$$964$$ 18086.2 0.604272
$$965$$ −15553.4 −0.518842
$$966$$ 0 0
$$967$$ 14464.0 0.481004 0.240502 0.970649i $$-0.422688\pi$$
0.240502 + 0.970649i $$0.422688\pi$$
$$968$$ 7448.00 0.247301
$$969$$ 0 0
$$970$$ −12320.0 −0.407806
$$971$$ −37832.9 −1.25038 −0.625188 0.780474i $$-0.714978\pi$$
−0.625188 + 0.780474i $$0.714978\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −27872.0 −0.916916
$$975$$ 0 0
$$976$$ 4352.71 0.142753
$$977$$ −42062.0 −1.37736 −0.688681 0.725065i $$-0.741810\pi$$
−0.688681 + 0.725065i $$0.741810\pi$$
$$978$$ 0 0
$$979$$ −30018.7 −0.979980
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −24552.0 −0.797847
$$983$$ 43020.5 1.39587 0.697935 0.716161i $$-0.254102\pi$$
0.697935 + 0.716161i $$0.254102\pi$$
$$984$$ 0 0
$$985$$ −9174.45 −0.296774
$$986$$ 18686.6 0.603553
$$987$$ 0 0
$$988$$ 2464.00 0.0793424
$$989$$ −20928.0 −0.672873
$$990$$ 0 0
$$991$$ 21272.0 0.681864 0.340932 0.940088i $$-0.389257\pi$$
0.340932 + 0.940088i $$0.389257\pi$$
$$992$$ 6604.11 0.211372
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 46288.0 1.47480
$$996$$ 0 0
$$997$$ 121.951 0.00387384 0.00193692 0.999998i $$-0.499383\pi$$
0.00193692 + 0.999998i $$0.499383\pi$$
$$998$$ 4440.00 0.140827
$$999$$ 0 0
Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000

## Twists

By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 882.4.a.w.1.1 2
3.2 odd 2 98.4.a.h.1.1 2
7.2 even 3 882.4.g.bi.361.2 4
7.3 odd 6 882.4.g.bi.667.1 4
7.4 even 3 882.4.g.bi.667.2 4
7.5 odd 6 882.4.g.bi.361.1 4
7.6 odd 2 inner 882.4.a.w.1.2 2
12.11 even 2 784.4.a.z.1.2 2
15.14 odd 2 2450.4.a.bs.1.2 2
21.2 odd 6 98.4.c.g.67.2 4
21.5 even 6 98.4.c.g.67.1 4
21.11 odd 6 98.4.c.g.79.2 4
21.17 even 6 98.4.c.g.79.1 4
21.20 even 2 98.4.a.h.1.2 yes 2
84.83 odd 2 784.4.a.z.1.1 2
105.104 even 2 2450.4.a.bs.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
98.4.a.h.1.1 2 3.2 odd 2
98.4.a.h.1.2 yes 2 21.20 even 2
98.4.c.g.67.1 4 21.5 even 6
98.4.c.g.67.2 4 21.2 odd 6
98.4.c.g.79.1 4 21.17 even 6
98.4.c.g.79.2 4 21.11 odd 6
784.4.a.z.1.1 2 84.83 odd 2
784.4.a.z.1.2 2 12.11 even 2
882.4.a.w.1.1 2 1.1 even 1 trivial
882.4.a.w.1.2 2 7.6 odd 2 inner
882.4.g.bi.361.1 4 7.5 odd 6
882.4.g.bi.361.2 4 7.2 even 3
882.4.g.bi.667.1 4 7.3 odd 6
882.4.g.bi.667.2 4 7.4 even 3
2450.4.a.bs.1.1 2 105.104 even 2
2450.4.a.bs.1.2 2 15.14 odd 2