# Properties

 Label 3450.2.a.p.1.1 Level $3450$ Weight $2$ Character 3450.1 Self dual yes Analytic conductor $27.548$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3450.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$27.5483886973$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 690) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 3450.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{8} +1.00000 q^{9} -4.00000 q^{11} -1.00000 q^{12} +6.00000 q^{13} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} +4.00000 q^{19} -4.00000 q^{22} -1.00000 q^{23} -1.00000 q^{24} +6.00000 q^{26} -1.00000 q^{27} -6.00000 q^{29} -8.00000 q^{31} +1.00000 q^{32} +4.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} -6.00000 q^{37} +4.00000 q^{38} -6.00000 q^{39} +10.0000 q^{41} -4.00000 q^{43} -4.00000 q^{44} -1.00000 q^{46} +8.00000 q^{47} -1.00000 q^{48} -7.00000 q^{49} -6.00000 q^{51} +6.00000 q^{52} +14.0000 q^{53} -1.00000 q^{54} -4.00000 q^{57} -6.00000 q^{58} +10.0000 q^{61} -8.00000 q^{62} +1.00000 q^{64} +4.00000 q^{66} -4.00000 q^{67} +6.00000 q^{68} +1.00000 q^{69} +8.00000 q^{71} +1.00000 q^{72} -2.00000 q^{73} -6.00000 q^{74} +4.00000 q^{76} -6.00000 q^{78} -12.0000 q^{79} +1.00000 q^{81} +10.0000 q^{82} +16.0000 q^{83} -4.00000 q^{86} +6.00000 q^{87} -4.00000 q^{88} -2.00000 q^{89} -1.00000 q^{92} +8.00000 q^{93} +8.00000 q^{94} -1.00000 q^{96} +14.0000 q^{97} -7.00000 q^{98} -4.00000 q^{99} +O(q^{100})$$

## 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 $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 6.00000 1.66410 0.832050 0.554700i $$-0.187167\pi$$
0.832050 + 0.554700i $$0.187167\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ 6.00000 1.45521 0.727607 0.685994i $$-0.240633\pi$$
0.727607 + 0.685994i $$0.240633\pi$$
$$18$$ 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
$$23$$ −1.00000 −0.208514
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ 6.00000 1.17670
$$27$$ −1.00000 −0.192450
$$28$$ 0 0
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 0 0
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 4.00000 0.696311
$$34$$ 6.00000 1.02899
$$35$$ 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
$$39$$ −6.00000 −0.960769
$$40$$ 0 0
$$41$$ 10.0000 1.56174 0.780869 0.624695i $$-0.214777\pi$$
0.780869 + 0.624695i $$0.214777\pi$$
$$42$$ 0 0
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 0 0
$$46$$ −1.00000 −0.147442
$$47$$ 8.00000 1.16692 0.583460 0.812142i $$-0.301699\pi$$
0.583460 + 0.812142i $$0.301699\pi$$
$$48$$ −1.00000 −0.144338
$$49$$ −7.00000 −1.00000
$$50$$ 0 0
$$51$$ −6.00000 −0.840168
$$52$$ 6.00000 0.832050
$$53$$ 14.0000 1.92305 0.961524 0.274721i $$-0.0885855\pi$$
0.961524 + 0.274721i $$0.0885855\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −4.00000 −0.529813
$$58$$ −6.00000 −0.787839
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ 10.0000 1.28037 0.640184 0.768221i $$-0.278858\pi$$
0.640184 + 0.768221i $$0.278858\pi$$
$$62$$ −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 $$0.578571\pi$$
$$68$$ 6.00000 0.727607
$$69$$ 1.00000 0.120386
$$70$$ 0 0
$$71$$ 8.00000 0.949425 0.474713 0.880141i $$-0.342552\pi$$
0.474713 + 0.880141i $$0.342552\pi$$
$$72$$ 1.00000 0.117851
$$73$$ −2.00000 −0.234082 −0.117041 0.993127i $$-0.537341\pi$$
−0.117041 + 0.993127i $$0.537341\pi$$
$$74$$ −6.00000 −0.697486
$$75$$ 0 0
$$76$$ 4.00000 0.458831
$$77$$ 0 0
$$78$$ −6.00000 −0.679366
$$79$$ −12.0000 −1.35011 −0.675053 0.737769i $$-0.735879\pi$$
−0.675053 + 0.737769i $$0.735879\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 10.0000 1.10432
$$83$$ 16.0000 1.75623 0.878114 0.478451i $$-0.158802\pi$$
0.878114 + 0.478451i $$0.158802\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ 6.00000 0.643268
$$88$$ −4.00000 −0.426401
$$89$$ −2.00000 −0.212000 −0.106000 0.994366i $$-0.533804\pi$$
−0.106000 + 0.994366i $$0.533804\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −1.00000 −0.104257
$$93$$ 8.00000 0.829561
$$94$$ 8.00000 0.825137
$$95$$ 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
$$101$$ 18.0000 1.79107 0.895533 0.444994i $$-0.146794\pi$$
0.895533 + 0.444994i $$0.146794\pi$$
$$102$$ −6.00000 −0.594089
$$103$$ 16.0000 1.57653 0.788263 0.615338i $$-0.210980\pi$$
0.788263 + 0.615338i $$0.210980\pi$$
$$104$$ 6.00000 0.588348
$$105$$ 0 0
$$106$$ 14.0000 1.35980
$$107$$ −8.00000 −0.773389 −0.386695 0.922208i $$-0.626383\pi$$
−0.386695 + 0.922208i $$0.626383\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 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
$$116$$ −6.00000 −0.557086
$$117$$ 6.00000 0.554700
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ 10.0000 0.905357
$$123$$ −10.0000 −0.901670
$$124$$ −8.00000 −0.718421
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −12.0000 −1.06483 −0.532414 0.846484i $$-0.678715\pi$$
−0.532414 + 0.846484i $$0.678715\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ 8.00000 0.698963 0.349482 0.936943i $$-0.386358\pi$$
0.349482 + 0.936943i $$0.386358\pi$$
$$132$$ 4.00000 0.348155
$$133$$ 0 0
$$134$$ −4.00000 −0.345547
$$135$$ 0 0
$$136$$ 6.00000 0.514496
$$137$$ −2.00000 −0.170872 −0.0854358 0.996344i $$-0.527228\pi$$
−0.0854358 + 0.996344i $$0.527228\pi$$
$$138$$ 1.00000 0.0851257
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ 0 0
$$141$$ −8.00000 −0.673722
$$142$$ 8.00000 0.671345
$$143$$ −24.0000 −2.00698
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ −2.00000 −0.165521
$$147$$ 7.00000 0.577350
$$148$$ −6.00000 −0.493197
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ 0 0
$$151$$ 16.0000 1.30206 0.651031 0.759051i $$-0.274337\pi$$
0.651031 + 0.759051i $$0.274337\pi$$
$$152$$ 4.00000 0.324443
$$153$$ 6.00000 0.485071
$$154$$ 0 0
$$155$$ 0 0
$$156$$ −6.00000 −0.480384
$$157$$ −6.00000 −0.478852 −0.239426 0.970915i $$-0.576959\pi$$
−0.239426 + 0.970915i $$0.576959\pi$$
$$158$$ −12.0000 −0.954669
$$159$$ −14.0000 −1.11027
$$160$$ 0 0
$$161$$ 0 0
$$162$$ 1.00000 0.0785674
$$163$$ −4.00000 −0.313304 −0.156652 0.987654i $$-0.550070\pi$$
−0.156652 + 0.987654i $$0.550070\pi$$
$$164$$ 10.0000 0.780869
$$165$$ 0 0
$$166$$ 16.0000 1.24184
$$167$$ 16.0000 1.23812 0.619059 0.785345i $$-0.287514\pi$$
0.619059 + 0.785345i $$0.287514\pi$$
$$168$$ 0 0
$$169$$ 23.0000 1.76923
$$170$$ 0 0
$$171$$ 4.00000 0.305888
$$172$$ −4.00000 −0.304997
$$173$$ −18.0000 −1.36851 −0.684257 0.729241i $$-0.739873\pi$$
−0.684257 + 0.729241i $$0.739873\pi$$
$$174$$ 6.00000 0.454859
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ 0 0
$$178$$ −2.00000 −0.149906
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ 0 0
$$181$$ −6.00000 −0.445976 −0.222988 0.974821i $$-0.571581\pi$$
−0.222988 + 0.974821i $$0.571581\pi$$
$$182$$ 0 0
$$183$$ −10.0000 −0.739221
$$184$$ −1.00000 −0.0737210
$$185$$ 0 0
$$186$$ 8.00000 0.586588
$$187$$ −24.0000 −1.75505
$$188$$ 8.00000 0.583460
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ −1.00000 −0.0721688
$$193$$ −10.0000 −0.719816 −0.359908 0.932988i $$-0.617192\pi$$
−0.359908 + 0.932988i $$0.617192\pi$$
$$194$$ 14.0000 1.00514
$$195$$ 0 0
$$196$$ −7.00000 −0.500000
$$197$$ −26.0000 −1.85242 −0.926212 0.377004i $$-0.876954\pi$$
−0.926212 + 0.377004i $$0.876954\pi$$
$$198$$ −4.00000 −0.284268
$$199$$ −12.0000 −0.850657 −0.425329 0.905039i $$-0.639842\pi$$
−0.425329 + 0.905039i $$0.639842\pi$$
$$200$$ 0 0
$$201$$ 4.00000 0.282138
$$202$$ 18.0000 1.26648
$$203$$ 0 0
$$204$$ −6.00000 −0.420084
$$205$$ 0 0
$$206$$ 16.0000 1.11477
$$207$$ −1.00000 −0.0695048
$$208$$ 6.00000 0.416025
$$209$$ −16.0000 −1.10674
$$210$$ 0 0
$$211$$ −20.0000 −1.37686 −0.688428 0.725304i $$-0.741699\pi$$
−0.688428 + 0.725304i $$0.741699\pi$$
$$212$$ 14.0000 0.961524
$$213$$ −8.00000 −0.548151
$$214$$ −8.00000 −0.546869
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ 0 0
$$218$$ 2.00000 0.135457
$$219$$ 2.00000 0.135147
$$220$$ 0 0
$$221$$ 36.0000 2.42162
$$222$$ 6.00000 0.402694
$$223$$ 20.0000 1.33930 0.669650 0.742677i $$-0.266444\pi$$
0.669650 + 0.742677i $$0.266444\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 14.0000 0.931266
$$227$$ −8.00000 −0.530979 −0.265489 0.964114i $$-0.585534\pi$$
−0.265489 + 0.964114i $$0.585534\pi$$
$$228$$ −4.00000 −0.264906
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −6.00000 −0.393919
$$233$$ −22.0000 −1.44127 −0.720634 0.693316i $$-0.756149\pi$$
−0.720634 + 0.693316i $$0.756149\pi$$
$$234$$ 6.00000 0.392232
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 12.0000 0.779484
$$238$$ 0 0
$$239$$ −8.00000 −0.517477 −0.258738 0.965947i $$-0.583307\pi$$
−0.258738 + 0.965947i $$0.583307\pi$$
$$240$$ 0 0
$$241$$ 2.00000 0.128831 0.0644157 0.997923i $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ 5.00000 0.321412
$$243$$ −1.00000 −0.0641500
$$244$$ 10.0000 0.640184
$$245$$ 0 0
$$246$$ −10.0000 −0.637577
$$247$$ 24.0000 1.52708
$$248$$ −8.00000 −0.508001
$$249$$ −16.0000 −1.01396
$$250$$ 0 0
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 0 0
$$253$$ 4.00000 0.251478
$$254$$ −12.0000 −0.752947
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −6.00000 −0.374270 −0.187135 0.982334i $$-0.559920\pi$$
−0.187135 + 0.982334i $$0.559920\pi$$
$$258$$ 4.00000 0.249029
$$259$$ 0 0
$$260$$ 0 0
$$261$$ −6.00000 −0.371391
$$262$$ 8.00000 0.494242
$$263$$ 32.0000 1.97320 0.986602 0.163144i $$-0.0521635\pi$$
0.986602 + 0.163144i $$0.0521635\pi$$
$$264$$ 4.00000 0.246183
$$265$$ 0 0
$$266$$ 0 0
$$267$$ 2.00000 0.122398
$$268$$ −4.00000 −0.244339
$$269$$ 10.0000 0.609711 0.304855 0.952399i $$-0.401392\pi$$
0.304855 + 0.952399i $$0.401392\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ 6.00000 0.363803
$$273$$ 0 0
$$274$$ −2.00000 −0.120824
$$275$$ 0 0
$$276$$ 1.00000 0.0601929
$$277$$ 30.0000 1.80253 0.901263 0.433273i $$-0.142641\pi$$
0.901263 + 0.433273i $$0.142641\pi$$
$$278$$ 12.0000 0.719712
$$279$$ −8.00000 −0.478947
$$280$$ 0 0
$$281$$ 30.0000 1.78965 0.894825 0.446417i $$-0.147300\pi$$
0.894825 + 0.446417i $$0.147300\pi$$
$$282$$ −8.00000 −0.476393
$$283$$ −4.00000 −0.237775 −0.118888 0.992908i $$-0.537933\pi$$
−0.118888 + 0.992908i $$0.537933\pi$$
$$284$$ 8.00000 0.474713
$$285$$ 0 0
$$286$$ −24.0000 −1.41915
$$287$$ 0 0
$$288$$ 1.00000 0.0589256
$$289$$ 19.0000 1.11765
$$290$$ 0 0
$$291$$ −14.0000 −0.820695
$$292$$ −2.00000 −0.117041
$$293$$ 14.0000 0.817889 0.408944 0.912559i $$-0.365897\pi$$
0.408944 + 0.912559i $$0.365897\pi$$
$$294$$ 7.00000 0.408248
$$295$$ 0 0
$$296$$ −6.00000 −0.348743
$$297$$ 4.00000 0.232104
$$298$$ 10.0000 0.579284
$$299$$ −6.00000 −0.346989
$$300$$ 0 0
$$301$$ 0 0
$$302$$ 16.0000 0.920697
$$303$$ −18.0000 −1.03407
$$304$$ 4.00000 0.229416
$$305$$ 0 0
$$306$$ 6.00000 0.342997
$$307$$ −12.0000 −0.684876 −0.342438 0.939540i $$-0.611253\pi$$
−0.342438 + 0.939540i $$0.611253\pi$$
$$308$$ 0 0
$$309$$ −16.0000 −0.910208
$$310$$ 0 0
$$311$$ −8.00000 −0.453638 −0.226819 0.973937i $$-0.572833\pi$$
−0.226819 + 0.973937i $$0.572833\pi$$
$$312$$ −6.00000 −0.339683
$$313$$ −10.0000 −0.565233 −0.282617 0.959233i $$-0.591202\pi$$
−0.282617 + 0.959233i $$0.591202\pi$$
$$314$$ −6.00000 −0.338600
$$315$$ 0 0
$$316$$ −12.0000 −0.675053
$$317$$ −18.0000 −1.01098 −0.505490 0.862832i $$-0.668688\pi$$
−0.505490 + 0.862832i $$0.668688\pi$$
$$318$$ −14.0000 −0.785081
$$319$$ 24.0000 1.34374
$$320$$ 0 0
$$321$$ 8.00000 0.446516
$$322$$ 0 0
$$323$$ 24.0000 1.33540
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ −4.00000 −0.221540
$$327$$ −2.00000 −0.110600
$$328$$ 10.0000 0.552158
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ 16.0000 0.878114
$$333$$ −6.00000 −0.328798
$$334$$ 16.0000 0.875481
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 14.0000 0.762629 0.381314 0.924445i $$-0.375472\pi$$
0.381314 + 0.924445i $$0.375472\pi$$
$$338$$ 23.0000 1.25104
$$339$$ −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
$$346$$ −18.0000 −0.967686
$$347$$ 28.0000 1.50312 0.751559 0.659665i $$-0.229302\pi$$
0.751559 + 0.659665i $$0.229302\pi$$
$$348$$ 6.00000 0.321634
$$349$$ −34.0000 −1.81998 −0.909989 0.414632i $$-0.863910\pi$$
−0.909989 + 0.414632i $$0.863910\pi$$
$$350$$ 0 0
$$351$$ −6.00000 −0.320256
$$352$$ −4.00000 −0.213201
$$353$$ −6.00000 −0.319348 −0.159674 0.987170i $$-0.551044\pi$$
−0.159674 + 0.987170i $$0.551044\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −2.00000 −0.106000
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −8.00000 −0.422224 −0.211112 0.977462i $$-0.567708\pi$$
−0.211112 + 0.977462i $$0.567708\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ −6.00000 −0.315353
$$363$$ −5.00000 −0.262432
$$364$$ 0 0
$$365$$ 0 0
$$366$$ −10.0000 −0.522708
$$367$$ 32.0000 1.67039 0.835193 0.549957i $$-0.185356\pi$$
0.835193 + 0.549957i $$0.185356\pi$$
$$368$$ −1.00000 −0.0521286
$$369$$ 10.0000 0.520579
$$370$$ 0 0
$$371$$ 0 0
$$372$$ 8.00000 0.414781
$$373$$ 18.0000 0.932005 0.466002 0.884783i $$-0.345694\pi$$
0.466002 + 0.884783i $$0.345694\pi$$
$$374$$ −24.0000 −1.24101
$$375$$ 0 0
$$376$$ 8.00000 0.412568
$$377$$ −36.0000 −1.85409
$$378$$ 0 0
$$379$$ 4.00000 0.205466 0.102733 0.994709i $$-0.467241\pi$$
0.102733 + 0.994709i $$0.467241\pi$$
$$380$$ 0 0
$$381$$ 12.0000 0.614779
$$382$$ −8.00000 −0.409316
$$383$$ −16.0000 −0.817562 −0.408781 0.912633i $$-0.634046\pi$$
−0.408781 + 0.912633i $$0.634046\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ −10.0000 −0.508987
$$387$$ −4.00000 −0.203331
$$388$$ 14.0000 0.710742
$$389$$ −6.00000 −0.304212 −0.152106 0.988364i $$-0.548606\pi$$
−0.152106 + 0.988364i $$0.548606\pi$$
$$390$$ 0 0
$$391$$ −6.00000 −0.303433
$$392$$ −7.00000 −0.353553
$$393$$ −8.00000 −0.403547
$$394$$ −26.0000 −1.30986
$$395$$ 0 0
$$396$$ −4.00000 −0.201008
$$397$$ −2.00000 −0.100377 −0.0501886 0.998740i $$-0.515982\pi$$
−0.0501886 + 0.998740i $$0.515982\pi$$
$$398$$ −12.0000 −0.601506
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 6.00000 0.299626 0.149813 0.988714i $$-0.452133\pi$$
0.149813 + 0.988714i $$0.452133\pi$$
$$402$$ 4.00000 0.199502
$$403$$ −48.0000 −2.39105
$$404$$ 18.0000 0.895533
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 24.0000 1.18964
$$408$$ −6.00000 −0.297044
$$409$$ 10.0000 0.494468 0.247234 0.968956i $$-0.420478\pi$$
0.247234 + 0.968956i $$0.420478\pi$$
$$410$$ 0 0
$$411$$ 2.00000 0.0986527
$$412$$ 16.0000 0.788263
$$413$$ 0 0
$$414$$ −1.00000 −0.0491473
$$415$$ 0 0
$$416$$ 6.00000 0.294174
$$417$$ −12.0000 −0.587643
$$418$$ −16.0000 −0.782586
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 0 0
$$421$$ −38.0000 −1.85201 −0.926003 0.377515i $$-0.876779\pi$$
−0.926003 + 0.377515i $$0.876779\pi$$
$$422$$ −20.0000 −0.973585
$$423$$ 8.00000 0.388973
$$424$$ 14.0000 0.679900
$$425$$ 0 0
$$426$$ −8.00000 −0.387601
$$427$$ 0 0
$$428$$ −8.00000 −0.386695
$$429$$ 24.0000 1.15873
$$430$$ 0 0
$$431$$ 8.00000 0.385346 0.192673 0.981263i $$-0.438284\pi$$
0.192673 + 0.981263i $$0.438284\pi$$
$$432$$ −1.00000 −0.0481125
$$433$$ −2.00000 −0.0961139 −0.0480569 0.998845i $$-0.515303\pi$$
−0.0480569 + 0.998845i $$0.515303\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 2.00000 0.0957826
$$437$$ −4.00000 −0.191346
$$438$$ 2.00000 0.0955637
$$439$$ −24.0000 −1.14546 −0.572729 0.819745i $$-0.694115\pi$$
−0.572729 + 0.819745i $$0.694115\pi$$
$$440$$ 0 0
$$441$$ −7.00000 −0.333333
$$442$$ 36.0000 1.71235
$$443$$ −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
$$446$$ 20.0000 0.947027
$$447$$ −10.0000 −0.472984
$$448$$ 0 0
$$449$$ 10.0000 0.471929 0.235965 0.971762i $$-0.424175\pi$$
0.235965 + 0.971762i $$0.424175\pi$$
$$450$$ 0 0
$$451$$ −40.0000 −1.88353
$$452$$ 14.0000 0.658505
$$453$$ −16.0000 −0.751746
$$454$$ −8.00000 −0.375459
$$455$$ 0 0
$$456$$ −4.00000 −0.187317
$$457$$ −10.0000 −0.467780 −0.233890 0.972263i $$-0.575146\pi$$
−0.233890 + 0.972263i $$0.575146\pi$$
$$458$$ −14.0000 −0.654177
$$459$$ −6.00000 −0.280056
$$460$$ 0 0
$$461$$ 2.00000 0.0931493 0.0465746 0.998915i $$-0.485169\pi$$
0.0465746 + 0.998915i $$0.485169\pi$$
$$462$$ 0 0
$$463$$ 20.0000 0.929479 0.464739 0.885448i $$-0.346148\pi$$
0.464739 + 0.885448i $$0.346148\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ −22.0000 −1.01913
$$467$$ −8.00000 −0.370196 −0.185098 0.982720i $$-0.559260\pi$$
−0.185098 + 0.982720i $$0.559260\pi$$
$$468$$ 6.00000 0.277350
$$469$$ 0 0
$$470$$ 0 0
$$471$$ 6.00000 0.276465
$$472$$ 0 0
$$473$$ 16.0000 0.735681
$$474$$ 12.0000 0.551178
$$475$$ 0 0
$$476$$ 0 0
$$477$$ 14.0000 0.641016
$$478$$ −8.00000 −0.365911
$$479$$ 24.0000 1.09659 0.548294 0.836286i $$-0.315277\pi$$
0.548294 + 0.836286i $$0.315277\pi$$
$$480$$ 0 0
$$481$$ −36.0000 −1.64146
$$482$$ 2.00000 0.0910975
$$483$$ 0 0
$$484$$ 5.00000 0.227273
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ −20.0000 −0.906287 −0.453143 0.891438i $$-0.649697\pi$$
−0.453143 + 0.891438i $$0.649697\pi$$
$$488$$ 10.0000 0.452679
$$489$$ 4.00000 0.180886
$$490$$ 0 0
$$491$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ −10.0000 −0.450835
$$493$$ −36.0000 −1.62136
$$494$$ 24.0000 1.07981
$$495$$ 0 0
$$496$$ −8.00000 −0.359211
$$497$$ 0 0
$$498$$ −16.0000 −0.716977
$$499$$ 12.0000 0.537194 0.268597 0.963253i $$-0.413440\pi$$
0.268597 + 0.963253i $$0.413440\pi$$
$$500$$ 0 0
$$501$$ −16.0000 −0.714827
$$502$$ −12.0000 −0.535586
$$503$$ 24.0000 1.07011 0.535054 0.844818i $$-0.320291\pi$$
0.535054 + 0.844818i $$0.320291\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 4.00000 0.177822
$$507$$ −23.0000 −1.02147
$$508$$ −12.0000 −0.532414
$$509$$ −6.00000 −0.265945 −0.132973 0.991120i $$-0.542452\pi$$
−0.132973 + 0.991120i $$0.542452\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ −4.00000 −0.176604
$$514$$ −6.00000 −0.264649
$$515$$ 0 0
$$516$$ 4.00000 0.176090
$$517$$ −32.0000 −1.40736
$$518$$ 0 0
$$519$$ 18.0000 0.790112
$$520$$ 0 0
$$521$$ −42.0000 −1.84005 −0.920027 0.391856i $$-0.871833\pi$$
−0.920027 + 0.391856i $$0.871833\pi$$
$$522$$ −6.00000 −0.262613
$$523$$ 20.0000 0.874539 0.437269 0.899331i $$-0.355946\pi$$
0.437269 + 0.899331i $$0.355946\pi$$
$$524$$ 8.00000 0.349482
$$525$$ 0 0
$$526$$ 32.0000 1.39527
$$527$$ −48.0000 −2.09091
$$528$$ 4.00000 0.174078
$$529$$ 1.00000 0.0434783
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 60.0000 2.59889
$$534$$ 2.00000 0.0865485
$$535$$ 0 0
$$536$$ −4.00000 −0.172774
$$537$$ 0 0
$$538$$ 10.0000 0.431131
$$539$$ 28.0000 1.20605
$$540$$ 0 0
$$541$$ −34.0000 −1.46177 −0.730887 0.682498i $$-0.760893\pi$$
−0.730887 + 0.682498i $$0.760893\pi$$
$$542$$ −8.00000 −0.343629
$$543$$ 6.00000 0.257485
$$544$$ 6.00000 0.257248
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −20.0000 −0.855138 −0.427569 0.903983i $$-0.640630\pi$$
−0.427569 + 0.903983i $$0.640630\pi$$
$$548$$ −2.00000 −0.0854358
$$549$$ 10.0000 0.426790
$$550$$ 0 0
$$551$$ −24.0000 −1.02243
$$552$$ 1.00000 0.0425628
$$553$$ 0 0
$$554$$ 30.0000 1.27458
$$555$$ 0 0
$$556$$ 12.0000 0.508913
$$557$$ −18.0000 −0.762684 −0.381342 0.924434i $$-0.624538\pi$$
−0.381342 + 0.924434i $$0.624538\pi$$
$$558$$ −8.00000 −0.338667
$$559$$ −24.0000 −1.01509
$$560$$ 0 0
$$561$$ 24.0000 1.01328
$$562$$ 30.0000 1.26547
$$563$$ 8.00000 0.337160 0.168580 0.985688i $$-0.446082\pi$$
0.168580 + 0.985688i $$0.446082\pi$$
$$564$$ −8.00000 −0.336861
$$565$$ 0 0
$$566$$ −4.00000 −0.168133
$$567$$ 0 0
$$568$$ 8.00000 0.335673
$$569$$ −18.0000 −0.754599 −0.377300 0.926091i $$-0.623147\pi$$
−0.377300 + 0.926091i $$0.623147\pi$$
$$570$$ 0 0
$$571$$ 20.0000 0.836974 0.418487 0.908223i $$-0.362561\pi$$
0.418487 + 0.908223i $$0.362561\pi$$
$$572$$ −24.0000 −1.00349
$$573$$ 8.00000 0.334205
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ −34.0000 −1.41544 −0.707719 0.706494i $$-0.750276\pi$$
−0.707719 + 0.706494i $$0.750276\pi$$
$$578$$ 19.0000 0.790296
$$579$$ 10.0000 0.415586
$$580$$ 0 0
$$581$$ 0 0
$$582$$ −14.0000 −0.580319
$$583$$ −56.0000 −2.31928
$$584$$ −2.00000 −0.0827606
$$585$$ 0 0
$$586$$ 14.0000 0.578335
$$587$$ 4.00000 0.165098 0.0825488 0.996587i $$-0.473694\pi$$
0.0825488 + 0.996587i $$0.473694\pi$$
$$588$$ 7.00000 0.288675
$$589$$ −32.0000 −1.31854
$$590$$ 0 0
$$591$$ 26.0000 1.06950
$$592$$ −6.00000 −0.246598
$$593$$ −6.00000 −0.246390 −0.123195 0.992382i $$-0.539314\pi$$
−0.123195 + 0.992382i $$0.539314\pi$$
$$594$$ 4.00000 0.164122
$$595$$ 0 0
$$596$$ 10.0000 0.409616
$$597$$ 12.0000 0.491127
$$598$$ −6.00000 −0.245358
$$599$$ −40.0000 −1.63436 −0.817178 0.576386i $$-0.804463\pi$$
−0.817178 + 0.576386i $$0.804463\pi$$
$$600$$ 0 0
$$601$$ 10.0000 0.407909 0.203954 0.978980i $$-0.434621\pi$$
0.203954 + 0.978980i $$0.434621\pi$$
$$602$$ 0 0
$$603$$ −4.00000 −0.162893
$$604$$ 16.0000 0.651031
$$605$$ 0 0
$$606$$ −18.0000 −0.731200
$$607$$ 28.0000 1.13648 0.568242 0.822861i $$-0.307624\pi$$
0.568242 + 0.822861i $$0.307624\pi$$
$$608$$ 4.00000 0.162221
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 48.0000 1.94187
$$612$$ 6.00000 0.242536
$$613$$ −14.0000 −0.565455 −0.282727 0.959200i $$-0.591239\pi$$
−0.282727 + 0.959200i $$0.591239\pi$$
$$614$$ −12.0000 −0.484281
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −18.0000 −0.724653 −0.362326 0.932051i $$-0.618017\pi$$
−0.362326 + 0.932051i $$0.618017\pi$$
$$618$$ −16.0000 −0.643614
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ 0 0
$$621$$ 1.00000 0.0401286
$$622$$ −8.00000 −0.320771
$$623$$ 0 0
$$624$$ −6.00000 −0.240192
$$625$$ 0 0
$$626$$ −10.0000 −0.399680
$$627$$ 16.0000 0.638978
$$628$$ −6.00000 −0.239426
$$629$$ −36.0000 −1.43541
$$630$$ 0 0
$$631$$ 20.0000 0.796187 0.398094 0.917345i $$-0.369672\pi$$
0.398094 + 0.917345i $$0.369672\pi$$
$$632$$ −12.0000 −0.477334
$$633$$ 20.0000 0.794929
$$634$$ −18.0000 −0.714871
$$635$$ 0 0
$$636$$ −14.0000 −0.555136
$$637$$ −42.0000 −1.66410
$$638$$ 24.0000 0.950169
$$639$$ 8.00000 0.316475
$$640$$ 0 0
$$641$$ −2.00000 −0.0789953 −0.0394976 0.999220i $$-0.512576\pi$$
−0.0394976 + 0.999220i $$0.512576\pi$$
$$642$$ 8.00000 0.315735
$$643$$ 44.0000 1.73519 0.867595 0.497271i $$-0.165665\pi$$
0.867595 + 0.497271i $$0.165665\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 24.0000 0.944267
$$647$$ −24.0000 −0.943537 −0.471769 0.881722i $$-0.656384\pi$$
−0.471769 + 0.881722i $$0.656384\pi$$
$$648$$ 1.00000 0.0392837
$$649$$ 0 0
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −4.00000 −0.156652
$$653$$ 14.0000 0.547862 0.273931 0.961749i $$-0.411676\pi$$
0.273931 + 0.961749i $$0.411676\pi$$
$$654$$ −2.00000 −0.0782062
$$655$$ 0 0
$$656$$ 10.0000 0.390434
$$657$$ −2.00000 −0.0780274
$$658$$ 0 0
$$659$$ −36.0000 −1.40236 −0.701180 0.712984i $$-0.747343\pi$$
−0.701180 + 0.712984i $$0.747343\pi$$
$$660$$ 0 0
$$661$$ 34.0000 1.32245 0.661223 0.750189i $$-0.270038\pi$$
0.661223 + 0.750189i $$0.270038\pi$$
$$662$$ −20.0000 −0.777322
$$663$$ −36.0000 −1.39812
$$664$$ 16.0000 0.620920
$$665$$ 0 0
$$666$$ −6.00000 −0.232495
$$667$$ 6.00000 0.232321
$$668$$ 16.0000 0.619059
$$669$$ −20.0000 −0.773245
$$670$$ 0 0
$$671$$ −40.0000 −1.54418
$$672$$ 0 0
$$673$$ 22.0000 0.848038 0.424019 0.905653i $$-0.360619\pi$$
0.424019 + 0.905653i $$0.360619\pi$$
$$674$$ 14.0000 0.539260
$$675$$ 0 0
$$676$$ 23.0000 0.884615
$$677$$ 6.00000 0.230599 0.115299 0.993331i $$-0.463217\pi$$
0.115299 + 0.993331i $$0.463217\pi$$
$$678$$ −14.0000 −0.537667
$$679$$ 0 0
$$680$$ 0 0
$$681$$ 8.00000 0.306561
$$682$$ 32.0000 1.22534
$$683$$ −44.0000 −1.68361 −0.841807 0.539779i $$-0.818508\pi$$
−0.841807 + 0.539779i $$0.818508\pi$$
$$684$$ 4.00000 0.152944
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 14.0000 0.534133
$$688$$ −4.00000 −0.152499
$$689$$ 84.0000 3.20015
$$690$$ 0 0
$$691$$ −28.0000 −1.06517 −0.532585 0.846376i $$-0.678779\pi$$
−0.532585 + 0.846376i $$0.678779\pi$$
$$692$$ −18.0000 −0.684257
$$693$$ 0 0
$$694$$ 28.0000 1.06287
$$695$$ 0 0
$$696$$ 6.00000 0.227429
$$697$$ 60.0000 2.27266
$$698$$ −34.0000 −1.28692
$$699$$ 22.0000 0.832116
$$700$$ 0 0
$$701$$ 18.0000 0.679851 0.339925 0.940452i $$-0.389598\pi$$
0.339925 + 0.940452i $$0.389598\pi$$
$$702$$ −6.00000 −0.226455
$$703$$ −24.0000 −0.905177
$$704$$ −4.00000 −0.150756
$$705$$ 0 0
$$706$$ −6.00000 −0.225813
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ 0 0
$$711$$ −12.0000 −0.450035
$$712$$ −2.00000 −0.0749532
$$713$$ 8.00000 0.299602
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 8.00000 0.298765
$$718$$ −8.00000 −0.298557
$$719$$ −40.0000 −1.49175 −0.745874 0.666087i $$-0.767968\pi$$
−0.745874 + 0.666087i $$0.767968\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −3.00000 −0.111648
$$723$$ −2.00000 −0.0743808
$$724$$ −6.00000 −0.222988
$$725$$ 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
$$731$$ −24.0000 −0.887672
$$732$$ −10.0000 −0.369611
$$733$$ −6.00000 −0.221615 −0.110808 0.993842i $$-0.535344\pi$$
−0.110808 + 0.993842i $$0.535344\pi$$
$$734$$ 32.0000 1.18114
$$735$$ 0 0
$$736$$ −1.00000 −0.0368605
$$737$$ 16.0000 0.589368
$$738$$ 10.0000 0.368105
$$739$$ −12.0000 −0.441427 −0.220714 0.975339i $$-0.570839\pi$$
−0.220714 + 0.975339i $$0.570839\pi$$
$$740$$ 0 0
$$741$$ −24.0000 −0.881662
$$742$$ 0 0
$$743$$ 24.0000 0.880475 0.440237 0.897881i $$-0.354894\pi$$
0.440237 + 0.897881i $$0.354894\pi$$
$$744$$ 8.00000 0.293294
$$745$$ 0 0
$$746$$ 18.0000 0.659027
$$747$$ 16.0000 0.585409
$$748$$ −24.0000 −0.877527
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −28.0000 −1.02173 −0.510867 0.859660i $$-0.670676\pi$$
−0.510867 + 0.859660i $$0.670676\pi$$
$$752$$ 8.00000 0.291730
$$753$$ 12.0000 0.437304
$$754$$ −36.0000 −1.31104
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −54.0000 −1.96266 −0.981332 0.192323i $$-0.938398\pi$$
−0.981332 + 0.192323i $$0.938398\pi$$
$$758$$ 4.00000 0.145287
$$759$$ −4.00000 −0.145191
$$760$$ 0 0
$$761$$ 10.0000 0.362500 0.181250 0.983437i $$-0.441986\pi$$
0.181250 + 0.983437i $$0.441986\pi$$
$$762$$ 12.0000 0.434714
$$763$$ 0 0
$$764$$ −8.00000 −0.289430
$$765$$ 0 0
$$766$$ −16.0000 −0.578103
$$767$$ 0 0
$$768$$ −1.00000 −0.0360844
$$769$$ −6.00000 −0.216366 −0.108183 0.994131i $$-0.534503\pi$$
−0.108183 + 0.994131i $$0.534503\pi$$
$$770$$ 0 0
$$771$$ 6.00000 0.216085
$$772$$ −10.0000 −0.359908
$$773$$ −2.00000 −0.0719350 −0.0359675 0.999353i $$-0.511451\pi$$
−0.0359675 + 0.999353i $$0.511451\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ 0 0
$$776$$ 14.0000 0.502571
$$777$$ 0 0
$$778$$ −6.00000 −0.215110
$$779$$ 40.0000 1.43315
$$780$$ 0 0
$$781$$ −32.0000 −1.14505
$$782$$ −6.00000 −0.214560
$$783$$ 6.00000 0.214423
$$784$$ −7.00000 −0.250000
$$785$$ 0 0
$$786$$ −8.00000 −0.285351
$$787$$ −36.0000 −1.28326 −0.641631 0.767014i $$-0.721742\pi$$
−0.641631 + 0.767014i $$0.721742\pi$$
$$788$$ −26.0000 −0.926212
$$789$$ −32.0000 −1.13923
$$790$$ 0 0
$$791$$ 0 0
$$792$$ −4.00000 −0.142134
$$793$$ 60.0000 2.13066
$$794$$ −2.00000 −0.0709773
$$795$$ 0 0
$$796$$ −12.0000 −0.425329
$$797$$ −18.0000 −0.637593 −0.318796 0.947823i $$-0.603279\pi$$
−0.318796 + 0.947823i $$0.603279\pi$$
$$798$$ 0 0
$$799$$ 48.0000 1.69812
$$800$$ 0 0
$$801$$ −2.00000 −0.0706665
$$802$$ 6.00000 0.211867
$$803$$ 8.00000 0.282314
$$804$$ 4.00000 0.141069
$$805$$ 0 0
$$806$$ −48.0000 −1.69073
$$807$$ −10.0000 −0.352017
$$808$$ 18.0000 0.633238
$$809$$ −30.0000 −1.05474 −0.527372 0.849635i $$-0.676823\pi$$
−0.527372 + 0.849635i $$0.676823\pi$$
$$810$$ 0 0
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ 0 0
$$813$$ 8.00000 0.280572
$$814$$ 24.0000 0.841200
$$815$$ 0 0
$$816$$ −6.00000 −0.210042
$$817$$ −16.0000 −0.559769
$$818$$ 10.0000 0.349642
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −46.0000 −1.60541 −0.802706 0.596376i $$-0.796607\pi$$
−0.802706 + 0.596376i $$0.796607\pi$$
$$822$$ 2.00000 0.0697580
$$823$$ −20.0000 −0.697156 −0.348578 0.937280i $$-0.613335\pi$$
−0.348578 + 0.937280i $$0.613335\pi$$
$$824$$ 16.0000 0.557386
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 8.00000 0.278187 0.139094 0.990279i $$-0.455581\pi$$
0.139094 + 0.990279i $$0.455581\pi$$
$$828$$ −1.00000 −0.0347524
$$829$$ −18.0000 −0.625166 −0.312583 0.949890i $$-0.601194\pi$$
−0.312583 + 0.949890i $$0.601194\pi$$
$$830$$ 0 0
$$831$$ −30.0000 −1.04069
$$832$$ 6.00000 0.208013
$$833$$ −42.0000 −1.45521
$$834$$ −12.0000 −0.415526
$$835$$ 0 0
$$836$$ −16.0000 −0.553372
$$837$$ 8.00000 0.276520
$$838$$ −12.0000 −0.414533
$$839$$ −48.0000 −1.65714 −0.828572 0.559883i $$-0.810846\pi$$
−0.828572 + 0.559883i $$0.810846\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ −38.0000 −1.30957
$$843$$ −30.0000 −1.03325
$$844$$ −20.0000 −0.688428
$$845$$ 0 0
$$846$$ 8.00000 0.275046
$$847$$ 0 0
$$848$$ 14.0000 0.480762
$$849$$ 4.00000 0.137280
$$850$$ 0 0
$$851$$ 6.00000 0.205677
$$852$$ −8.00000 −0.274075
$$853$$ 22.0000 0.753266 0.376633 0.926363i $$-0.377082\pi$$
0.376633 + 0.926363i $$0.377082\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ −8.00000 −0.273434
$$857$$ 42.0000 1.43469 0.717346 0.696717i $$-0.245357\pi$$
0.717346 + 0.696717i $$0.245357\pi$$
$$858$$ 24.0000 0.819346
$$859$$ 4.00000 0.136478 0.0682391 0.997669i $$-0.478262\pi$$
0.0682391 + 0.997669i $$0.478262\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 8.00000 0.272481
$$863$$ −24.0000 −0.816970 −0.408485 0.912765i $$-0.633943\pi$$
−0.408485 + 0.912765i $$0.633943\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ −2.00000 −0.0679628
$$867$$ −19.0000 −0.645274
$$868$$ 0 0
$$869$$ 48.0000 1.62829
$$870$$ 0 0
$$871$$ −24.0000 −0.813209
$$872$$ 2.00000 0.0677285
$$873$$ 14.0000 0.473828
$$874$$ −4.00000 −0.135302
$$875$$ 0 0
$$876$$ 2.00000 0.0675737
$$877$$ −2.00000 −0.0675352 −0.0337676 0.999430i $$-0.510751\pi$$
−0.0337676 + 0.999430i $$0.510751\pi$$
$$878$$ −24.0000 −0.809961
$$879$$ −14.0000 −0.472208
$$880$$ 0 0
$$881$$ 6.00000 0.202145 0.101073 0.994879i $$-0.467773\pi$$
0.101073 + 0.994879i $$0.467773\pi$$
$$882$$ −7.00000 −0.235702
$$883$$ −44.0000 −1.48072 −0.740359 0.672212i $$-0.765344\pi$$
−0.740359 + 0.672212i $$0.765344\pi$$
$$884$$ 36.0000 1.21081
$$885$$ 0 0
$$886$$ −12.0000 −0.403148
$$887$$ −8.00000 −0.268614 −0.134307 0.990940i $$-0.542881\pi$$
−0.134307 + 0.990940i $$0.542881\pi$$
$$888$$ 6.00000 0.201347
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −4.00000 −0.134005
$$892$$ 20.0000 0.669650
$$893$$ 32.0000 1.07084
$$894$$ −10.0000 −0.334450
$$895$$ 0 0
$$896$$ 0 0
$$897$$ 6.00000 0.200334
$$898$$ 10.0000 0.333704
$$899$$ 48.0000 1.60089
$$900$$ 0 0
$$901$$ 84.0000 2.79845
$$902$$ −40.0000 −1.33185
$$903$$ 0 0
$$904$$ 14.0000 0.465633
$$905$$ 0 0
$$906$$ −16.0000 −0.531564
$$907$$ 4.00000 0.132818 0.0664089 0.997792i $$-0.478846\pi$$
0.0664089 + 0.997792i $$0.478846\pi$$
$$908$$ −8.00000 −0.265489
$$909$$ 18.0000 0.597022
$$910$$ 0 0
$$911$$ 8.00000 0.265052 0.132526 0.991180i $$-0.457691\pi$$
0.132526 + 0.991180i $$0.457691\pi$$
$$912$$ −4.00000 −0.132453
$$913$$ −64.0000 −2.11809
$$914$$ −10.0000 −0.330771
$$915$$ 0 0
$$916$$ −14.0000 −0.462573
$$917$$ 0 0
$$918$$ −6.00000 −0.198030
$$919$$ 12.0000 0.395843 0.197922 0.980218i $$-0.436581\pi$$
0.197922 + 0.980218i $$0.436581\pi$$
$$920$$ 0 0
$$921$$ 12.0000 0.395413
$$922$$ 2.00000 0.0658665
$$923$$ 48.0000 1.57994
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 20.0000 0.657241
$$927$$ 16.0000 0.525509
$$928$$ −6.00000 −0.196960
$$929$$ −6.00000 −0.196854 −0.0984268 0.995144i $$-0.531381\pi$$
−0.0984268 + 0.995144i $$0.531381\pi$$
$$930$$ 0 0
$$931$$ −28.0000 −0.917663
$$932$$ −22.0000 −0.720634
$$933$$ 8.00000 0.261908
$$934$$ −8.00000 −0.261768
$$935$$ 0 0
$$936$$ 6.00000 0.196116
$$937$$ −10.0000 −0.326686 −0.163343 0.986569i $$-0.552228\pi$$
−0.163343 + 0.986569i $$0.552228\pi$$
$$938$$ 0 0
$$939$$ 10.0000 0.326338
$$940$$ 0 0
$$941$$ −14.0000 −0.456387 −0.228193 0.973616i $$-0.573282\pi$$
−0.228193 + 0.973616i $$0.573282\pi$$
$$942$$ 6.00000 0.195491
$$943$$ −10.0000 −0.325645
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 16.0000 0.520205
$$947$$ 20.0000 0.649913 0.324956 0.945729i $$-0.394650\pi$$
0.324956 + 0.945729i $$0.394650\pi$$
$$948$$ 12.0000 0.389742
$$949$$ −12.0000 −0.389536
$$950$$ 0 0
$$951$$ 18.0000 0.583690
$$952$$ 0 0
$$953$$ 22.0000 0.712650 0.356325 0.934362i $$-0.384030\pi$$
0.356325 + 0.934362i $$0.384030\pi$$
$$954$$ 14.0000 0.453267
$$955$$ 0 0
$$956$$ −8.00000 −0.258738
$$957$$ −24.0000 −0.775810
$$958$$ 24.0000 0.775405
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ −36.0000 −1.16069
$$963$$ −8.00000 −0.257796
$$964$$ 2.00000 0.0644157
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −28.0000 −0.900419 −0.450210 0.892923i $$-0.648651\pi$$
−0.450210 + 0.892923i $$0.648651\pi$$
$$968$$ 5.00000 0.160706
$$969$$ −24.0000 −0.770991
$$970$$ 0 0
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ −1.00000 −0.0320750
$$973$$ 0 0
$$974$$ −20.0000 −0.640841
$$975$$ 0 0
$$976$$ 10.0000 0.320092
$$977$$ −2.00000 −0.0639857 −0.0319928 0.999488i $$-0.510185\pi$$
−0.0319928 + 0.999488i $$0.510185\pi$$
$$978$$ 4.00000 0.127906
$$979$$ 8.00000 0.255681
$$980$$ 0 0
$$981$$ 2.00000 0.0638551
$$982$$ 0 0
$$983$$ 24.0000 0.765481 0.382741 0.923856i $$-0.374980\pi$$
0.382741 + 0.923856i $$0.374980\pi$$
$$984$$ −10.0000 −0.318788
$$985$$ 0 0
$$986$$ −36.0000 −1.14647
$$987$$ 0 0
$$988$$ 24.0000 0.763542
$$989$$ 4.00000 0.127193
$$990$$ 0 0
$$991$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$992$$ −8.00000 −0.254000
$$993$$ 20.0000 0.634681
$$994$$ 0 0
$$995$$ 0 0
$$996$$ −16.0000 −0.506979
$$997$$ 22.0000 0.696747 0.348373 0.937356i $$-0.386734\pi$$
0.348373 + 0.937356i $$0.386734\pi$$
$$998$$ 12.0000 0.379853
$$999$$ 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 3450.2.a.p.1.1 1
5.2 odd 4 3450.2.d.a.2899.2 2
5.3 odd 4 3450.2.d.a.2899.1 2
5.4 even 2 690.2.a.e.1.1 1
15.14 odd 2 2070.2.a.r.1.1 1
20.19 odd 2 5520.2.a.f.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.e.1.1 1 5.4 even 2
2070.2.a.r.1.1 1 15.14 odd 2
3450.2.a.p.1.1 1 1.1 even 1 trivial
3450.2.d.a.2899.1 2 5.3 odd 4
3450.2.d.a.2899.2 2 5.2 odd 4
5520.2.a.f.1.1 1 20.19 odd 2