# Properties

 Label 2550.2.a.l.1.1 Level $2550$ Weight $2$ Character 2550.1 Self dual yes Analytic conductor $20.362$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2550 = 2 \cdot 3 \cdot 5^{2} \cdot 17$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2550.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$20.3618525154$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 510) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 2550.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +4.00000 q^{11} +1.00000 q^{12} +2.00000 q^{13} +1.00000 q^{16} -1.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} -4.00000 q^{22} -1.00000 q^{24} -2.00000 q^{26} +1.00000 q^{27} -2.00000 q^{29} +8.00000 q^{31} -1.00000 q^{32} +4.00000 q^{33} +1.00000 q^{34} +1.00000 q^{36} -6.00000 q^{37} -4.00000 q^{38} +2.00000 q^{39} -6.00000 q^{41} +4.00000 q^{43} +4.00000 q^{44} +1.00000 q^{48} -7.00000 q^{49} -1.00000 q^{51} +2.00000 q^{52} +10.0000 q^{53} -1.00000 q^{54} +4.00000 q^{57} +2.00000 q^{58} -4.00000 q^{59} -2.00000 q^{61} -8.00000 q^{62} +1.00000 q^{64} -4.00000 q^{66} -4.00000 q^{67} -1.00000 q^{68} -1.00000 q^{72} +6.00000 q^{73} +6.00000 q^{74} +4.00000 q^{76} -2.00000 q^{78} +8.00000 q^{79} +1.00000 q^{81} +6.00000 q^{82} +12.0000 q^{83} -4.00000 q^{86} -2.00000 q^{87} -4.00000 q^{88} -6.00000 q^{89} +8.00000 q^{93} -1.00000 q^{96} +14.0000 q^{97} +7.00000 q^{98} +4.00000 q^{99} +O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −1.00000 −0.707107
$$3$$ 1.00000 0.577350
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ −1.00000 −0.408248
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 0 0
$$11$$ 4.00000 1.20605 0.603023 0.797724i $$-0.293963\pi$$
0.603023 + 0.797724i $$0.293963\pi$$
$$12$$ 1.00000 0.288675
$$13$$ 2.00000 0.554700 0.277350 0.960769i $$-0.410544\pi$$
0.277350 + 0.960769i $$0.410544\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −1.00000 −0.242536
$$18$$ −1.00000 −0.235702
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −4.00000 −0.852803
$$23$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ −2.00000 −0.392232
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ 8.00000 1.43684 0.718421 0.695608i $$-0.244865\pi$$
0.718421 + 0.695608i $$0.244865\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 4.00000 0.696311
$$34$$ 1.00000 0.171499
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ −6.00000 −0.986394 −0.493197 0.869918i $$-0.664172\pi$$
−0.493197 + 0.869918i $$0.664172\pi$$
$$38$$ −4.00000 −0.648886
$$39$$ 2.00000 0.320256
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 4.00000 0.603023
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −7.00000 −1.00000
$$50$$ 0 0
$$51$$ −1.00000 −0.140028
$$52$$ 2.00000 0.277350
$$53$$ 10.0000 1.37361 0.686803 0.726844i $$-0.259014\pi$$
0.686803 + 0.726844i $$0.259014\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 0 0
$$57$$ 4.00000 0.529813
$$58$$ 2.00000 0.262613
$$59$$ −4.00000 −0.520756 −0.260378 0.965507i $$-0.583847\pi$$
−0.260378 + 0.965507i $$0.583847\pi$$
$$60$$ 0 0
$$61$$ −2.00000 −0.256074 −0.128037 0.991769i $$-0.540868\pi$$
−0.128037 + 0.991769i $$0.540868\pi$$
$$62$$ −8.00000 −1.01600
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ −4.00000 −0.492366
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ −1.00000 −0.121268
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ −1.00000 −0.117851
$$73$$ 6.00000 0.702247 0.351123 0.936329i $$-0.385800\pi$$
0.351123 + 0.936329i $$0.385800\pi$$
$$74$$ 6.00000 0.697486
$$75$$ 0 0
$$76$$ 4.00000 0.458831
$$77$$ 0 0
$$78$$ −2.00000 −0.226455
$$79$$ 8.00000 0.900070 0.450035 0.893011i $$-0.351411\pi$$
0.450035 + 0.893011i $$0.351411\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 6.00000 0.662589
$$83$$ 12.0000 1.31717 0.658586 0.752506i $$-0.271155\pi$$
0.658586 + 0.752506i $$0.271155\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ −2.00000 −0.214423
$$88$$ −4.00000 −0.426401
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ 0 0
$$93$$ 8.00000 0.829561
$$94$$ 0 0
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ 14.0000 1.42148 0.710742 0.703452i $$-0.248359\pi$$
0.710742 + 0.703452i $$0.248359\pi$$
$$98$$ 7.00000 0.707107
$$99$$ 4.00000 0.402015
$$100$$ 0 0
$$101$$ −10.0000 −0.995037 −0.497519 0.867453i $$-0.665755\pi$$
−0.497519 + 0.867453i $$0.665755\pi$$
$$102$$ 1.00000 0.0990148
$$103$$ 8.00000 0.788263 0.394132 0.919054i $$-0.371045\pi$$
0.394132 + 0.919054i $$0.371045\pi$$
$$104$$ −2.00000 −0.196116
$$105$$ 0 0
$$106$$ −10.0000 −0.971286
$$107$$ −4.00000 −0.386695 −0.193347 0.981130i $$-0.561934\pi$$
−0.193347 + 0.981130i $$0.561934\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ 14.0000 1.34096 0.670478 0.741929i $$-0.266089\pi$$
0.670478 + 0.741929i $$0.266089\pi$$
$$110$$ 0 0
$$111$$ −6.00000 −0.569495
$$112$$ 0 0
$$113$$ 14.0000 1.31701 0.658505 0.752577i $$-0.271189\pi$$
0.658505 + 0.752577i $$0.271189\pi$$
$$114$$ −4.00000 −0.374634
$$115$$ 0 0
$$116$$ −2.00000 −0.185695
$$117$$ 2.00000 0.184900
$$118$$ 4.00000 0.368230
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 2.00000 0.181071
$$123$$ −6.00000 −0.541002
$$124$$ 8.00000 0.718421
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ −20.0000 −1.74741 −0.873704 0.486458i $$-0.838289\pi$$
−0.873704 + 0.486458i $$0.838289\pi$$
$$132$$ 4.00000 0.348155
$$133$$ 0 0
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ 1.00000 0.0857493
$$137$$ −10.0000 −0.854358 −0.427179 0.904167i $$-0.640493\pi$$
−0.427179 + 0.904167i $$0.640493\pi$$
$$138$$ 0 0
$$139$$ 20.0000 1.69638 0.848189 0.529694i $$-0.177693\pi$$
0.848189 + 0.529694i $$0.177693\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ 8.00000 0.668994
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −6.00000 −0.496564
$$147$$ −7.00000 −0.577350
$$148$$ −6.00000 −0.493197
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ −4.00000 −0.324443
$$153$$ −1.00000 −0.0808452
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 2.00000 0.160128
$$157$$ 18.0000 1.43656 0.718278 0.695756i $$-0.244931\pi$$
0.718278 + 0.695756i $$0.244931\pi$$
$$158$$ −8.00000 −0.636446
$$159$$ 10.0000 0.793052
$$160$$ 0 0
$$161$$ 0 0
$$162$$ −1.00000 −0.0785674
$$163$$ −12.0000 −0.939913 −0.469956 0.882690i $$-0.655730\pi$$
−0.469956 + 0.882690i $$0.655730\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ −12.0000 −0.931381
$$167$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$168$$ 0 0
$$169$$ −9.00000 −0.692308
$$170$$ 0 0
$$171$$ 4.00000 0.305888
$$172$$ 4.00000 0.304997
$$173$$ −14.0000 −1.06440 −0.532200 0.846619i $$-0.678635\pi$$
−0.532200 + 0.846619i $$0.678635\pi$$
$$174$$ 2.00000 0.151620
$$175$$ 0 0
$$176$$ 4.00000 0.301511
$$177$$ −4.00000 −0.300658
$$178$$ 6.00000 0.449719
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ 22.0000 1.63525 0.817624 0.575753i $$-0.195291\pi$$
0.817624 + 0.575753i $$0.195291\pi$$
$$182$$ 0 0
$$183$$ −2.00000 −0.147844
$$184$$ 0 0
$$185$$ 0 0
$$186$$ −8.00000 −0.586588
$$187$$ −4.00000 −0.292509
$$188$$ 0 0
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ −18.0000 −1.29567 −0.647834 0.761781i $$-0.724325\pi$$
−0.647834 + 0.761781i $$0.724325\pi$$
$$194$$ −14.0000 −1.00514
$$195$$ 0 0
$$196$$ −7.00000 −0.500000
$$197$$ −6.00000 −0.427482 −0.213741 0.976890i $$-0.568565\pi$$
−0.213741 + 0.976890i $$0.568565\pi$$
$$198$$ −4.00000 −0.284268
$$199$$ −16.0000 −1.13421 −0.567105 0.823646i $$-0.691937\pi$$
−0.567105 + 0.823646i $$0.691937\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ 10.0000 0.703598
$$203$$ 0 0
$$204$$ −1.00000 −0.0700140
$$205$$ 0 0
$$206$$ −8.00000 −0.557386
$$207$$ 0 0
$$208$$ 2.00000 0.138675
$$209$$ 16.0000 1.10674
$$210$$ 0 0
$$211$$ 12.0000 0.826114 0.413057 0.910705i $$-0.364461\pi$$
0.413057 + 0.910705i $$0.364461\pi$$
$$212$$ 10.0000 0.686803
$$213$$ 0 0
$$214$$ 4.00000 0.273434
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ 0 0
$$218$$ −14.0000 −0.948200
$$219$$ 6.00000 0.405442
$$220$$ 0 0
$$221$$ −2.00000 −0.134535
$$222$$ 6.00000 0.402694
$$223$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ −14.0000 −0.931266
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ 4.00000 0.264906
$$229$$ 6.00000 0.396491 0.198246 0.980152i $$-0.436476\pi$$
0.198246 + 0.980152i $$0.436476\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 2.00000 0.131306
$$233$$ 22.0000 1.44127 0.720634 0.693316i $$-0.243851\pi$$
0.720634 + 0.693316i $$0.243851\pi$$
$$234$$ −2.00000 −0.130744
$$235$$ 0 0
$$236$$ −4.00000 −0.260378
$$237$$ 8.00000 0.519656
$$238$$ 0 0
$$239$$ 16.0000 1.03495 0.517477 0.855697i $$-0.326871\pi$$
0.517477 + 0.855697i $$0.326871\pi$$
$$240$$ 0 0
$$241$$ 18.0000 1.15948 0.579741 0.814801i $$-0.303154\pi$$
0.579741 + 0.814801i $$0.303154\pi$$
$$242$$ −5.00000 −0.321412
$$243$$ 1.00000 0.0641500
$$244$$ −2.00000 −0.128037
$$245$$ 0 0
$$246$$ 6.00000 0.382546
$$247$$ 8.00000 0.509028
$$248$$ −8.00000 −0.508001
$$249$$ 12.0000 0.760469
$$250$$ 0 0
$$251$$ 12.0000 0.757433 0.378717 0.925513i $$-0.376365\pi$$
0.378717 + 0.925513i $$0.376365\pi$$
$$252$$ 0 0
$$253$$ 0 0
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −2.00000 −0.124757 −0.0623783 0.998053i $$-0.519869\pi$$
−0.0623783 + 0.998053i $$0.519869\pi$$
$$258$$ −4.00000 −0.249029
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −2.00000 −0.123797
$$262$$ 20.0000 1.23560
$$263$$ 24.0000 1.47990 0.739952 0.672660i $$-0.234848\pi$$
0.739952 + 0.672660i $$0.234848\pi$$
$$264$$ −4.00000 −0.246183
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −6.00000 −0.367194
$$268$$ −4.00000 −0.244339
$$269$$ −18.0000 −1.09748 −0.548740 0.835993i $$-0.684892\pi$$
−0.548740 + 0.835993i $$0.684892\pi$$
$$270$$ 0 0
$$271$$ −16.0000 −0.971931 −0.485965 0.873978i $$-0.661532\pi$$
−0.485965 + 0.873978i $$0.661532\pi$$
$$272$$ −1.00000 −0.0606339
$$273$$ 0 0
$$274$$ 10.0000 0.604122
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 10.0000 0.600842 0.300421 0.953807i $$-0.402873\pi$$
0.300421 + 0.953807i $$0.402873\pi$$
$$278$$ −20.0000 −1.19952
$$279$$ 8.00000 0.478947
$$280$$ 0 0
$$281$$ −6.00000 −0.357930 −0.178965 0.983855i $$-0.557275\pi$$
−0.178965 + 0.983855i $$0.557275\pi$$
$$282$$ 0 0
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ −8.00000 −0.473050
$$287$$ 0 0
$$288$$ −1.00000 −0.0589256
$$289$$ 1.00000 0.0588235
$$290$$ 0 0
$$291$$ 14.0000 0.820695
$$292$$ 6.00000 0.351123
$$293$$ −6.00000 −0.350524 −0.175262 0.984522i $$-0.556077\pi$$
−0.175262 + 0.984522i $$0.556077\pi$$
$$294$$ 7.00000 0.408248
$$295$$ 0 0
$$296$$ 6.00000 0.348743
$$297$$ 4.00000 0.232104
$$298$$ −6.00000 −0.347571
$$299$$ 0 0
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −8.00000 −0.460348
$$303$$ −10.0000 −0.574485
$$304$$ 4.00000 0.229416
$$305$$ 0 0
$$306$$ 1.00000 0.0571662
$$307$$ −20.0000 −1.14146 −0.570730 0.821138i $$-0.693340\pi$$
−0.570730 + 0.821138i $$0.693340\pi$$
$$308$$ 0 0
$$309$$ 8.00000 0.455104
$$310$$ 0 0
$$311$$ 16.0000 0.907277 0.453638 0.891186i $$-0.350126\pi$$
0.453638 + 0.891186i $$0.350126\pi$$
$$312$$ −2.00000 −0.113228
$$313$$ 22.0000 1.24351 0.621757 0.783210i $$-0.286419\pi$$
0.621757 + 0.783210i $$0.286419\pi$$
$$314$$ −18.0000 −1.01580
$$315$$ 0 0
$$316$$ 8.00000 0.450035
$$317$$ −30.0000 −1.68497 −0.842484 0.538721i $$-0.818908\pi$$
−0.842484 + 0.538721i $$0.818908\pi$$
$$318$$ −10.0000 −0.560772
$$319$$ −8.00000 −0.447914
$$320$$ 0 0
$$321$$ −4.00000 −0.223258
$$322$$ 0 0
$$323$$ −4.00000 −0.222566
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 12.0000 0.664619
$$327$$ 14.0000 0.774202
$$328$$ 6.00000 0.331295
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ 12.0000 0.658586
$$333$$ −6.00000 −0.328798
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 30.0000 1.63420 0.817102 0.576493i $$-0.195579\pi$$
0.817102 + 0.576493i $$0.195579\pi$$
$$338$$ 9.00000 0.489535
$$339$$ 14.0000 0.760376
$$340$$ 0 0
$$341$$ 32.0000 1.73290
$$342$$ −4.00000 −0.216295
$$343$$ 0 0
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ 14.0000 0.752645
$$347$$ −4.00000 −0.214731 −0.107366 0.994220i $$-0.534242\pi$$
−0.107366 + 0.994220i $$0.534242\pi$$
$$348$$ −2.00000 −0.107211
$$349$$ −2.00000 −0.107058 −0.0535288 0.998566i $$-0.517047\pi$$
−0.0535288 + 0.998566i $$0.517047\pi$$
$$350$$ 0 0
$$351$$ 2.00000 0.106752
$$352$$ −4.00000 −0.213201
$$353$$ −34.0000 −1.80964 −0.904819 0.425797i $$-0.859994\pi$$
−0.904819 + 0.425797i $$0.859994\pi$$
$$354$$ 4.00000 0.212598
$$355$$ 0 0
$$356$$ −6.00000 −0.317999
$$357$$ 0 0
$$358$$ 12.0000 0.634220
$$359$$ 24.0000 1.26667 0.633336 0.773877i $$-0.281685\pi$$
0.633336 + 0.773877i $$0.281685\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ −22.0000 −1.15629
$$363$$ 5.00000 0.262432
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 2.00000 0.104542
$$367$$ −8.00000 −0.417597 −0.208798 0.977959i $$-0.566955\pi$$
−0.208798 + 0.977959i $$0.566955\pi$$
$$368$$ 0 0
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 8.00000 0.414781
$$373$$ −6.00000 −0.310668 −0.155334 0.987862i $$-0.549645\pi$$
−0.155334 + 0.987862i $$0.549645\pi$$
$$374$$ 4.00000 0.206835
$$375$$ 0 0
$$376$$ 0 0
$$377$$ −4.00000 −0.206010
$$378$$ 0 0
$$379$$ 4.00000 0.205466 0.102733 0.994709i $$-0.467241\pi$$
0.102733 + 0.994709i $$0.467241\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 32.0000 1.63512 0.817562 0.575841i $$-0.195325\pi$$
0.817562 + 0.575841i $$0.195325\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 18.0000 0.916176
$$387$$ 4.00000 0.203331
$$388$$ 14.0000 0.710742
$$389$$ 22.0000 1.11544 0.557722 0.830028i $$-0.311675\pi$$
0.557722 + 0.830028i $$0.311675\pi$$
$$390$$ 0 0
$$391$$ 0 0
$$392$$ 7.00000 0.353553
$$393$$ −20.0000 −1.00887
$$394$$ 6.00000 0.302276
$$395$$ 0 0
$$396$$ 4.00000 0.201008
$$397$$ 18.0000 0.903394 0.451697 0.892171i $$-0.350819\pi$$
0.451697 + 0.892171i $$0.350819\pi$$
$$398$$ 16.0000 0.802008
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −30.0000 −1.49813 −0.749064 0.662497i $$-0.769497\pi$$
−0.749064 + 0.662497i $$0.769497\pi$$
$$402$$ 4.00000 0.199502
$$403$$ 16.0000 0.797017
$$404$$ −10.0000 −0.497519
$$405$$ 0 0
$$406$$ 0 0
$$407$$ −24.0000 −1.18964
$$408$$ 1.00000 0.0495074
$$409$$ −38.0000 −1.87898 −0.939490 0.342578i $$-0.888700\pi$$
−0.939490 + 0.342578i $$0.888700\pi$$
$$410$$ 0 0
$$411$$ −10.0000 −0.493264
$$412$$ 8.00000 0.394132
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −2.00000 −0.0980581
$$417$$ 20.0000 0.979404
$$418$$ −16.0000 −0.782586
$$419$$ −36.0000 −1.75872 −0.879358 0.476162i $$-0.842028\pi$$
−0.879358 + 0.476162i $$0.842028\pi$$
$$420$$ 0 0
$$421$$ 6.00000 0.292422 0.146211 0.989253i $$-0.453292\pi$$
0.146211 + 0.989253i $$0.453292\pi$$
$$422$$ −12.0000 −0.584151
$$423$$ 0 0
$$424$$ −10.0000 −0.485643
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 0 0
$$428$$ −4.00000 −0.193347
$$429$$ 8.00000 0.386244
$$430$$ 0 0
$$431$$ 24.0000 1.15604 0.578020 0.816023i $$-0.303826\pi$$
0.578020 + 0.816023i $$0.303826\pi$$
$$432$$ 1.00000 0.0481125
$$433$$ −18.0000 −0.865025 −0.432512 0.901628i $$-0.642373\pi$$
−0.432512 + 0.901628i $$0.642373\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 14.0000 0.670478
$$437$$ 0 0
$$438$$ −6.00000 −0.286691
$$439$$ −32.0000 −1.52728 −0.763638 0.645644i $$-0.776589\pi$$
−0.763638 + 0.645644i $$0.776589\pi$$
$$440$$ 0 0
$$441$$ −7.00000 −0.333333
$$442$$ 2.00000 0.0951303
$$443$$ −12.0000 −0.570137 −0.285069 0.958507i $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ −6.00000 −0.284747
$$445$$ 0 0
$$446$$ 0 0
$$447$$ 6.00000 0.283790
$$448$$ 0 0
$$449$$ 18.0000 0.849473 0.424736 0.905317i $$-0.360367\pi$$
0.424736 + 0.905317i $$0.360367\pi$$
$$450$$ 0 0
$$451$$ −24.0000 −1.13012
$$452$$ 14.0000 0.658505
$$453$$ 8.00000 0.375873
$$454$$ 12.0000 0.563188
$$455$$ 0 0
$$456$$ −4.00000 −0.187317
$$457$$ 22.0000 1.02912 0.514558 0.857455i $$-0.327956\pi$$
0.514558 + 0.857455i $$0.327956\pi$$
$$458$$ −6.00000 −0.280362
$$459$$ −1.00000 −0.0466760
$$460$$ 0 0
$$461$$ 30.0000 1.39724 0.698620 0.715493i $$-0.253798\pi$$
0.698620 + 0.715493i $$0.253798\pi$$
$$462$$ 0 0
$$463$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ 0 0
$$466$$ −22.0000 −1.01913
$$467$$ 12.0000 0.555294 0.277647 0.960683i $$-0.410445\pi$$
0.277647 + 0.960683i $$0.410445\pi$$
$$468$$ 2.00000 0.0924500
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 18.0000 0.829396
$$472$$ 4.00000 0.184115
$$473$$ 16.0000 0.735681
$$474$$ −8.00000 −0.367452
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 10.0000 0.457869
$$478$$ −16.0000 −0.731823
$$479$$ −40.0000 −1.82765 −0.913823 0.406112i $$-0.866884\pi$$
−0.913823 + 0.406112i $$0.866884\pi$$
$$480$$ 0 0
$$481$$ −12.0000 −0.547153
$$482$$ −18.0000 −0.819878
$$483$$ 0 0
$$484$$ 5.00000 0.227273
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ −32.0000 −1.45006 −0.725029 0.688718i $$-0.758174\pi$$
−0.725029 + 0.688718i $$0.758174\pi$$
$$488$$ 2.00000 0.0905357
$$489$$ −12.0000 −0.542659
$$490$$ 0 0
$$491$$ −20.0000 −0.902587 −0.451294 0.892375i $$-0.649037\pi$$
−0.451294 + 0.892375i $$0.649037\pi$$
$$492$$ −6.00000 −0.270501
$$493$$ 2.00000 0.0900755
$$494$$ −8.00000 −0.359937
$$495$$ 0 0
$$496$$ 8.00000 0.359211
$$497$$ 0 0
$$498$$ −12.0000 −0.537733
$$499$$ −20.0000 −0.895323 −0.447661 0.894203i $$-0.647743\pi$$
−0.447661 + 0.894203i $$0.647743\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −12.0000 −0.535586
$$503$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −9.00000 −0.399704
$$508$$ 0 0
$$509$$ −18.0000 −0.797836 −0.398918 0.916987i $$-0.630614\pi$$
−0.398918 + 0.916987i $$0.630614\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ 4.00000 0.176604
$$514$$ 2.00000 0.0882162
$$515$$ 0 0
$$516$$ 4.00000 0.176090
$$517$$ 0 0
$$518$$ 0 0
$$519$$ −14.0000 −0.614532
$$520$$ 0 0
$$521$$ −6.00000 −0.262865 −0.131432 0.991325i $$-0.541958\pi$$
−0.131432 + 0.991325i $$0.541958\pi$$
$$522$$ 2.00000 0.0875376
$$523$$ −44.0000 −1.92399 −0.961993 0.273075i $$-0.911959\pi$$
−0.961993 + 0.273075i $$0.911959\pi$$
$$524$$ −20.0000 −0.873704
$$525$$ 0 0
$$526$$ −24.0000 −1.04645
$$527$$ −8.00000 −0.348485
$$528$$ 4.00000 0.174078
$$529$$ −23.0000 −1.00000
$$530$$ 0 0
$$531$$ −4.00000 −0.173585
$$532$$ 0 0
$$533$$ −12.0000 −0.519778
$$534$$ 6.00000 0.259645
$$535$$ 0 0
$$536$$ 4.00000 0.172774
$$537$$ −12.0000 −0.517838
$$538$$ 18.0000 0.776035
$$539$$ −28.0000 −1.20605
$$540$$ 0 0
$$541$$ −2.00000 −0.0859867 −0.0429934 0.999075i $$-0.513689\pi$$
−0.0429934 + 0.999075i $$0.513689\pi$$
$$542$$ 16.0000 0.687259
$$543$$ 22.0000 0.944110
$$544$$ 1.00000 0.0428746
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −28.0000 −1.19719 −0.598597 0.801050i $$-0.704275\pi$$
−0.598597 + 0.801050i $$0.704275\pi$$
$$548$$ −10.0000 −0.427179
$$549$$ −2.00000 −0.0853579
$$550$$ 0 0
$$551$$ −8.00000 −0.340811
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −10.0000 −0.424859
$$555$$ 0 0
$$556$$ 20.0000 0.848189
$$557$$ −14.0000 −0.593199 −0.296600 0.955002i $$-0.595853\pi$$
−0.296600 + 0.955002i $$0.595853\pi$$
$$558$$ −8.00000 −0.338667
$$559$$ 8.00000 0.338364
$$560$$ 0 0
$$561$$ −4.00000 −0.168880
$$562$$ 6.00000 0.253095
$$563$$ 44.0000 1.85438 0.927189 0.374593i $$-0.122217\pi$$
0.927189 + 0.374593i $$0.122217\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 4.00000 0.168133
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ −44.0000 −1.84134 −0.920671 0.390339i $$-0.872358\pi$$
−0.920671 + 0.390339i $$0.872358\pi$$
$$572$$ 8.00000 0.334497
$$573$$ 0 0
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ 30.0000 1.24892 0.624458 0.781058i $$-0.285320\pi$$
0.624458 + 0.781058i $$0.285320\pi$$
$$578$$ −1.00000 −0.0415945
$$579$$ −18.0000 −0.748054
$$580$$ 0 0
$$581$$ 0 0
$$582$$ −14.0000 −0.580319
$$583$$ 40.0000 1.65663
$$584$$ −6.00000 −0.248282
$$585$$ 0 0
$$586$$ 6.00000 0.247858
$$587$$ −12.0000 −0.495293 −0.247647 0.968850i $$-0.579657\pi$$
−0.247647 + 0.968850i $$0.579657\pi$$
$$588$$ −7.00000 −0.288675
$$589$$ 32.0000 1.31854
$$590$$ 0 0
$$591$$ −6.00000 −0.246807
$$592$$ −6.00000 −0.246598
$$593$$ −18.0000 −0.739171 −0.369586 0.929197i $$-0.620500\pi$$
−0.369586 + 0.929197i $$0.620500\pi$$
$$594$$ −4.00000 −0.164122
$$595$$ 0 0
$$596$$ 6.00000 0.245770
$$597$$ −16.0000 −0.654836
$$598$$ 0 0
$$599$$ −40.0000 −1.63436 −0.817178 0.576386i $$-0.804463\pi$$
−0.817178 + 0.576386i $$0.804463\pi$$
$$600$$ 0 0
$$601$$ 26.0000 1.06056 0.530281 0.847822i $$-0.322086\pi$$
0.530281 + 0.847822i $$0.322086\pi$$
$$602$$ 0 0
$$603$$ −4.00000 −0.162893
$$604$$ 8.00000 0.325515
$$605$$ 0 0
$$606$$ 10.0000 0.406222
$$607$$ −8.00000 −0.324710 −0.162355 0.986732i $$-0.551909\pi$$
−0.162355 + 0.986732i $$0.551909\pi$$
$$608$$ −4.00000 −0.162221
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 0 0
$$612$$ −1.00000 −0.0404226
$$613$$ −22.0000 −0.888572 −0.444286 0.895885i $$-0.646543\pi$$
−0.444286 + 0.895885i $$0.646543\pi$$
$$614$$ 20.0000 0.807134
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 22.0000 0.885687 0.442843 0.896599i $$-0.353970\pi$$
0.442843 + 0.896599i $$0.353970\pi$$
$$618$$ −8.00000 −0.321807
$$619$$ 4.00000 0.160774 0.0803868 0.996764i $$-0.474384\pi$$
0.0803868 + 0.996764i $$0.474384\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −16.0000 −0.641542
$$623$$ 0 0
$$624$$ 2.00000 0.0800641
$$625$$ 0 0
$$626$$ −22.0000 −0.879297
$$627$$ 16.0000 0.638978
$$628$$ 18.0000 0.718278
$$629$$ 6.00000 0.239236
$$630$$ 0 0
$$631$$ 40.0000 1.59237 0.796187 0.605050i $$-0.206847\pi$$
0.796187 + 0.605050i $$0.206847\pi$$
$$632$$ −8.00000 −0.318223
$$633$$ 12.0000 0.476957
$$634$$ 30.0000 1.19145
$$635$$ 0 0
$$636$$ 10.0000 0.396526
$$637$$ −14.0000 −0.554700
$$638$$ 8.00000 0.316723
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 18.0000 0.710957 0.355479 0.934684i $$-0.384318\pi$$
0.355479 + 0.934684i $$0.384318\pi$$
$$642$$ 4.00000 0.157867
$$643$$ −28.0000 −1.10421 −0.552106 0.833774i $$-0.686176\pi$$
−0.552106 + 0.833774i $$0.686176\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 4.00000 0.157378
$$647$$ −24.0000 −0.943537 −0.471769 0.881722i $$-0.656384\pi$$
−0.471769 + 0.881722i $$0.656384\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ −16.0000 −0.628055
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −12.0000 −0.469956
$$653$$ −14.0000 −0.547862 −0.273931 0.961749i $$-0.588324\pi$$
−0.273931 + 0.961749i $$0.588324\pi$$
$$654$$ −14.0000 −0.547443
$$655$$ 0 0
$$656$$ −6.00000 −0.234261
$$657$$ 6.00000 0.234082
$$658$$ 0 0
$$659$$ 4.00000 0.155818 0.0779089 0.996960i $$-0.475176\pi$$
0.0779089 + 0.996960i $$0.475176\pi$$
$$660$$ 0 0
$$661$$ −10.0000 −0.388955 −0.194477 0.980907i $$-0.562301\pi$$
−0.194477 + 0.980907i $$0.562301\pi$$
$$662$$ 4.00000 0.155464
$$663$$ −2.00000 −0.0776736
$$664$$ −12.0000 −0.465690
$$665$$ 0 0
$$666$$ 6.00000 0.232495
$$667$$ 0 0
$$668$$ 0 0
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −8.00000 −0.308837
$$672$$ 0 0
$$673$$ −50.0000 −1.92736 −0.963679 0.267063i $$-0.913947\pi$$
−0.963679 + 0.267063i $$0.913947\pi$$
$$674$$ −30.0000 −1.15556
$$675$$ 0 0
$$676$$ −9.00000 −0.346154
$$677$$ −6.00000 −0.230599 −0.115299 0.993331i $$-0.536783\pi$$
−0.115299 + 0.993331i $$0.536783\pi$$
$$678$$ −14.0000 −0.537667
$$679$$ 0 0
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ −32.0000 −1.22534
$$683$$ −4.00000 −0.153056 −0.0765279 0.997067i $$-0.524383\pi$$
−0.0765279 + 0.997067i $$0.524383\pi$$
$$684$$ 4.00000 0.152944
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 6.00000 0.228914
$$688$$ 4.00000 0.152499
$$689$$ 20.0000 0.761939
$$690$$ 0 0
$$691$$ 12.0000 0.456502 0.228251 0.973602i $$-0.426699\pi$$
0.228251 + 0.973602i $$0.426699\pi$$
$$692$$ −14.0000 −0.532200
$$693$$ 0 0
$$694$$ 4.00000 0.151838
$$695$$ 0 0
$$696$$ 2.00000 0.0758098
$$697$$ 6.00000 0.227266
$$698$$ 2.00000 0.0757011
$$699$$ 22.0000 0.832116
$$700$$ 0 0
$$701$$ −18.0000 −0.679851 −0.339925 0.940452i $$-0.610402\pi$$
−0.339925 + 0.940452i $$0.610402\pi$$
$$702$$ −2.00000 −0.0754851
$$703$$ −24.0000 −0.905177
$$704$$ 4.00000 0.150756
$$705$$ 0 0
$$706$$ 34.0000 1.27961
$$707$$ 0 0
$$708$$ −4.00000 −0.150329
$$709$$ −26.0000 −0.976450 −0.488225 0.872718i $$-0.662356\pi$$
−0.488225 + 0.872718i $$0.662356\pi$$
$$710$$ 0 0
$$711$$ 8.00000 0.300023
$$712$$ 6.00000 0.224860
$$713$$ 0 0
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ 16.0000 0.597531
$$718$$ −24.0000 −0.895672
$$719$$ −24.0000 −0.895049 −0.447524 0.894272i $$-0.647694\pi$$
−0.447524 + 0.894272i $$0.647694\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ 3.00000 0.111648
$$723$$ 18.0000 0.669427
$$724$$ 22.0000 0.817624
$$725$$ 0 0
$$726$$ −5.00000 −0.185567
$$727$$ −8.00000 −0.296704 −0.148352 0.988935i $$-0.547397\pi$$
−0.148352 + 0.988935i $$0.547397\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −4.00000 −0.147945
$$732$$ −2.00000 −0.0739221
$$733$$ −14.0000 −0.517102 −0.258551 0.965998i $$-0.583245\pi$$
−0.258551 + 0.965998i $$0.583245\pi$$
$$734$$ 8.00000 0.295285
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −16.0000 −0.589368
$$738$$ 6.00000 0.220863
$$739$$ 20.0000 0.735712 0.367856 0.929883i $$-0.380092\pi$$
0.367856 + 0.929883i $$0.380092\pi$$
$$740$$ 0 0
$$741$$ 8.00000 0.293887
$$742$$ 0 0
$$743$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$744$$ −8.00000 −0.293294
$$745$$ 0 0
$$746$$ 6.00000 0.219676
$$747$$ 12.0000 0.439057
$$748$$ −4.00000 −0.146254
$$749$$ 0 0
$$750$$ 0 0
$$751$$ 24.0000 0.875772 0.437886 0.899030i $$-0.355727\pi$$
0.437886 + 0.899030i $$0.355727\pi$$
$$752$$ 0 0
$$753$$ 12.0000 0.437304
$$754$$ 4.00000 0.145671
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −38.0000 −1.38113 −0.690567 0.723269i $$-0.742639\pi$$
−0.690567 + 0.723269i $$0.742639\pi$$
$$758$$ −4.00000 −0.145287
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −6.00000 −0.217500 −0.108750 0.994069i $$-0.534685\pi$$
−0.108750 + 0.994069i $$0.534685\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ 0 0
$$765$$ 0 0
$$766$$ −32.0000 −1.15621
$$767$$ −8.00000 −0.288863
$$768$$ 1.00000 0.0360844
$$769$$ 2.00000 0.0721218 0.0360609 0.999350i $$-0.488519\pi$$
0.0360609 + 0.999350i $$0.488519\pi$$
$$770$$ 0 0
$$771$$ −2.00000 −0.0720282
$$772$$ −18.0000 −0.647834
$$773$$ −38.0000 −1.36677 −0.683383 0.730061i $$-0.739492\pi$$
−0.683383 + 0.730061i $$0.739492\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ 0 0
$$776$$ −14.0000 −0.502571
$$777$$ 0 0
$$778$$ −22.0000 −0.788738
$$779$$ −24.0000 −0.859889
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 0 0
$$783$$ −2.00000 −0.0714742
$$784$$ −7.00000 −0.250000
$$785$$ 0 0
$$786$$ 20.0000 0.713376
$$787$$ −12.0000 −0.427754 −0.213877 0.976861i $$-0.568609\pi$$
−0.213877 + 0.976861i $$0.568609\pi$$
$$788$$ −6.00000 −0.213741
$$789$$ 24.0000 0.854423
$$790$$ 0 0
$$791$$ 0 0
$$792$$ −4.00000 −0.142134
$$793$$ −4.00000 −0.142044
$$794$$ −18.0000 −0.638796
$$795$$ 0 0
$$796$$ −16.0000 −0.567105
$$797$$ 2.00000 0.0708436 0.0354218 0.999372i $$-0.488723\pi$$
0.0354218 + 0.999372i $$0.488723\pi$$
$$798$$ 0 0
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ 30.0000 1.05934
$$803$$ 24.0000 0.846942
$$804$$ −4.00000 −0.141069
$$805$$ 0 0
$$806$$ −16.0000 −0.563576
$$807$$ −18.0000 −0.633630
$$808$$ 10.0000 0.351799
$$809$$ −6.00000 −0.210949 −0.105474 0.994422i $$-0.533636\pi$$
−0.105474 + 0.994422i $$0.533636\pi$$
$$810$$ 0 0
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ 0 0
$$813$$ −16.0000 −0.561144
$$814$$ 24.0000 0.841200
$$815$$ 0 0
$$816$$ −1.00000 −0.0350070
$$817$$ 16.0000 0.559769
$$818$$ 38.0000 1.32864
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 54.0000 1.88461 0.942306 0.334751i $$-0.108652\pi$$
0.942306 + 0.334751i $$0.108652\pi$$
$$822$$ 10.0000 0.348790
$$823$$ −16.0000 −0.557725 −0.278862 0.960331i $$-0.589957\pi$$
−0.278862 + 0.960331i $$0.589957\pi$$
$$824$$ −8.00000 −0.278693
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −4.00000 −0.139094 −0.0695468 0.997579i $$-0.522155\pi$$
−0.0695468 + 0.997579i $$0.522155\pi$$
$$828$$ 0 0
$$829$$ 30.0000 1.04194 0.520972 0.853574i $$-0.325570\pi$$
0.520972 + 0.853574i $$0.325570\pi$$
$$830$$ 0 0
$$831$$ 10.0000 0.346896
$$832$$ 2.00000 0.0693375
$$833$$ 7.00000 0.242536
$$834$$ −20.0000 −0.692543
$$835$$ 0 0
$$836$$ 16.0000 0.553372
$$837$$ 8.00000 0.276520
$$838$$ 36.0000 1.24360
$$839$$ 16.0000 0.552381 0.276191 0.961103i $$-0.410928\pi$$
0.276191 + 0.961103i $$0.410928\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ −6.00000 −0.206774
$$843$$ −6.00000 −0.206651
$$844$$ 12.0000 0.413057
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 10.0000 0.343401
$$849$$ −4.00000 −0.137280
$$850$$ 0 0
$$851$$ 0 0
$$852$$ 0 0
$$853$$ 10.0000 0.342393 0.171197 0.985237i $$-0.445237\pi$$
0.171197 + 0.985237i $$0.445237\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 4.00000 0.136717
$$857$$ 6.00000 0.204956 0.102478 0.994735i $$-0.467323\pi$$
0.102478 + 0.994735i $$0.467323\pi$$
$$858$$ −8.00000 −0.273115
$$859$$ −20.0000 −0.682391 −0.341196 0.939992i $$-0.610832\pi$$
−0.341196 + 0.939992i $$0.610832\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −24.0000 −0.817443
$$863$$ −48.0000 −1.63394 −0.816970 0.576681i $$-0.804348\pi$$
−0.816970 + 0.576681i $$0.804348\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ 18.0000 0.611665
$$867$$ 1.00000 0.0339618
$$868$$ 0 0
$$869$$ 32.0000 1.08553
$$870$$ 0 0
$$871$$ −8.00000 −0.271070
$$872$$ −14.0000 −0.474100
$$873$$ 14.0000 0.473828
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 6.00000 0.202721
$$877$$ −14.0000 −0.472746 −0.236373 0.971662i $$-0.575959\pi$$
−0.236373 + 0.971662i $$0.575959\pi$$
$$878$$ 32.0000 1.07995
$$879$$ −6.00000 −0.202375
$$880$$ 0 0
$$881$$ −30.0000 −1.01073 −0.505363 0.862907i $$-0.668641\pi$$
−0.505363 + 0.862907i $$0.668641\pi$$
$$882$$ 7.00000 0.235702
$$883$$ −52.0000 −1.74994 −0.874970 0.484178i $$-0.839119\pi$$
−0.874970 + 0.484178i $$0.839119\pi$$
$$884$$ −2.00000 −0.0672673
$$885$$ 0 0
$$886$$ 12.0000 0.403148
$$887$$ 48.0000 1.61168 0.805841 0.592132i $$-0.201714\pi$$
0.805841 + 0.592132i $$0.201714\pi$$
$$888$$ 6.00000 0.201347
$$889$$ 0 0
$$890$$ 0 0
$$891$$ 4.00000 0.134005
$$892$$ 0 0
$$893$$ 0 0
$$894$$ −6.00000 −0.200670
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −18.0000 −0.600668
$$899$$ −16.0000 −0.533630
$$900$$ 0 0
$$901$$ −10.0000 −0.333148
$$902$$ 24.0000 0.799113
$$903$$ 0 0
$$904$$ −14.0000 −0.465633
$$905$$ 0 0
$$906$$ −8.00000 −0.265782
$$907$$ 12.0000 0.398453 0.199227 0.979953i $$-0.436157\pi$$
0.199227 + 0.979953i $$0.436157\pi$$
$$908$$ −12.0000 −0.398234
$$909$$ −10.0000 −0.331679
$$910$$ 0 0
$$911$$ −40.0000 −1.32526 −0.662630 0.748947i $$-0.730560\pi$$
−0.662630 + 0.748947i $$0.730560\pi$$
$$912$$ 4.00000 0.132453
$$913$$ 48.0000 1.58857
$$914$$ −22.0000 −0.727695
$$915$$ 0 0
$$916$$ 6.00000 0.198246
$$917$$ 0 0
$$918$$ 1.00000 0.0330049
$$919$$ −40.0000 −1.31948 −0.659739 0.751495i $$-0.729333\pi$$
−0.659739 + 0.751495i $$0.729333\pi$$
$$920$$ 0 0
$$921$$ −20.0000 −0.659022
$$922$$ −30.0000 −0.987997
$$923$$ 0 0
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ 8.00000 0.262754
$$928$$ 2.00000 0.0656532
$$929$$ 18.0000 0.590561 0.295280 0.955411i $$-0.404587\pi$$
0.295280 + 0.955411i $$0.404587\pi$$
$$930$$ 0 0
$$931$$ −28.0000 −0.917663
$$932$$ 22.0000 0.720634
$$933$$ 16.0000 0.523816
$$934$$ −12.0000 −0.392652
$$935$$ 0 0
$$936$$ −2.00000 −0.0653720
$$937$$ −10.0000 −0.326686 −0.163343 0.986569i $$-0.552228\pi$$
−0.163343 + 0.986569i $$0.552228\pi$$
$$938$$ 0 0
$$939$$ 22.0000 0.717943
$$940$$ 0 0
$$941$$ 46.0000 1.49956 0.749779 0.661689i $$-0.230160\pi$$
0.749779 + 0.661689i $$0.230160\pi$$
$$942$$ −18.0000 −0.586472
$$943$$ 0 0
$$944$$ −4.00000 −0.130189
$$945$$ 0 0
$$946$$ −16.0000 −0.520205
$$947$$ 20.0000 0.649913 0.324956 0.945729i $$-0.394650\pi$$
0.324956 + 0.945729i $$0.394650\pi$$
$$948$$ 8.00000 0.259828
$$949$$ 12.0000 0.389536
$$950$$ 0 0
$$951$$ −30.0000 −0.972817
$$952$$ 0 0
$$953$$ −58.0000 −1.87880 −0.939402 0.342817i $$-0.888619\pi$$
−0.939402 + 0.342817i $$0.888619\pi$$
$$954$$ −10.0000 −0.323762
$$955$$ 0 0
$$956$$ 16.0000 0.517477
$$957$$ −8.00000 −0.258603
$$958$$ 40.0000 1.29234
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ 12.0000 0.386896
$$963$$ −4.00000 −0.128898
$$964$$ 18.0000 0.579741
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 24.0000 0.771788 0.385894 0.922543i $$-0.373893\pi$$
0.385894 + 0.922543i $$0.373893\pi$$
$$968$$ −5.00000 −0.160706
$$969$$ −4.00000 −0.128499
$$970$$ 0 0
$$971$$ −20.0000 −0.641831 −0.320915 0.947108i $$-0.603990\pi$$
−0.320915 + 0.947108i $$0.603990\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 0 0
$$974$$ 32.0000 1.02535
$$975$$ 0 0
$$976$$ −2.00000 −0.0640184
$$977$$ 46.0000 1.47167 0.735835 0.677161i $$-0.236790\pi$$
0.735835 + 0.677161i $$0.236790\pi$$
$$978$$ 12.0000 0.383718
$$979$$ −24.0000 −0.767043
$$980$$ 0 0
$$981$$ 14.0000 0.446986
$$982$$ 20.0000 0.638226
$$983$$ −48.0000 −1.53096 −0.765481 0.643458i $$-0.777499\pi$$
−0.765481 + 0.643458i $$0.777499\pi$$
$$984$$ 6.00000 0.191273
$$985$$ 0 0
$$986$$ −2.00000 −0.0636930
$$987$$ 0 0
$$988$$ 8.00000 0.254514
$$989$$ 0 0
$$990$$ 0 0
$$991$$ −56.0000 −1.77890 −0.889449 0.457034i $$-0.848912\pi$$
−0.889449 + 0.457034i $$0.848912\pi$$
$$992$$ −8.00000 −0.254000
$$993$$ −4.00000 −0.126936
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 12.0000 0.380235
$$997$$ −38.0000 −1.20347 −0.601736 0.798695i $$-0.705524\pi$$
−0.601736 + 0.798695i $$0.705524\pi$$
$$998$$ 20.0000 0.633089
$$999$$ −6.00000 −0.189832
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 2550.2.a.l.1.1 1
3.2 odd 2 7650.2.a.bx.1.1 1
5.2 odd 4 2550.2.d.k.2449.1 2
5.3 odd 4 2550.2.d.k.2449.2 2
5.4 even 2 510.2.a.e.1.1 1
15.14 odd 2 1530.2.a.b.1.1 1
20.19 odd 2 4080.2.a.ba.1.1 1
85.84 even 2 8670.2.a.v.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
510.2.a.e.1.1 1 5.4 even 2
1530.2.a.b.1.1 1 15.14 odd 2
2550.2.a.l.1.1 1 1.1 even 1 trivial
2550.2.d.k.2449.1 2 5.2 odd 4
2550.2.d.k.2449.2 2 5.3 odd 4
4080.2.a.ba.1.1 1 20.19 odd 2
7650.2.a.bx.1.1 1 3.2 odd 2
8670.2.a.v.1.1 1 85.84 even 2