# Properties

 Label 3150.2.a.bl.1.1 Level $3150$ Weight $2$ Character 3150.1 Self dual yes Analytic conductor $25.153$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$3150 = 2 \cdot 3^{2} \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 3150.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$25.1528766367$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 1050) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 3150.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} +1.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{7} +1.00000 q^{8} -2.00000 q^{11} -1.00000 q^{13} +1.00000 q^{14} +1.00000 q^{16} -1.00000 q^{17} +4.00000 q^{19} -2.00000 q^{22} +7.00000 q^{23} -1.00000 q^{26} +1.00000 q^{28} -1.00000 q^{29} +3.00000 q^{31} +1.00000 q^{32} -1.00000 q^{34} +6.00000 q^{37} +4.00000 q^{38} +3.00000 q^{41} +1.00000 q^{43} -2.00000 q^{44} +7.00000 q^{46} -12.0000 q^{47} +1.00000 q^{49} -1.00000 q^{52} +11.0000 q^{53} +1.00000 q^{56} -1.00000 q^{58} +3.00000 q^{59} +5.00000 q^{61} +3.00000 q^{62} +1.00000 q^{64} +12.0000 q^{67} -1.00000 q^{68} -4.00000 q^{71} +14.0000 q^{73} +6.00000 q^{74} +4.00000 q^{76} -2.00000 q^{77} -2.00000 q^{79} +3.00000 q^{82} -3.00000 q^{83} +1.00000 q^{86} -2.00000 q^{88} -10.0000 q^{89} -1.00000 q^{91} +7.00000 q^{92} -12.0000 q^{94} +10.0000 q^{97} +1.00000 q^{98} +O(q^{100})$$

## 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
$$3$$ 0 0
$$4$$ 1.00000 0.500000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 1.00000 0.377964
$$8$$ 1.00000 0.353553
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −2.00000 −0.603023 −0.301511 0.953463i $$-0.597491\pi$$
−0.301511 + 0.953463i $$0.597491\pi$$
$$12$$ 0 0
$$13$$ −1.00000 −0.277350 −0.138675 0.990338i $$-0.544284\pi$$
−0.138675 + 0.990338i $$0.544284\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −1.00000 −0.242536 −0.121268 0.992620i $$-0.538696\pi$$
−0.121268 + 0.992620i $$0.538696\pi$$
$$18$$ 0 0
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −2.00000 −0.426401
$$23$$ 7.00000 1.45960 0.729800 0.683660i $$-0.239613\pi$$
0.729800 + 0.683660i $$0.239613\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −1.00000 −0.196116
$$27$$ 0 0
$$28$$ 1.00000 0.188982
$$29$$ −1.00000 −0.185695 −0.0928477 0.995680i $$-0.529597\pi$$
−0.0928477 + 0.995680i $$0.529597\pi$$
$$30$$ 0 0
$$31$$ 3.00000 0.538816 0.269408 0.963026i $$-0.413172\pi$$
0.269408 + 0.963026i $$0.413172\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 0 0
$$34$$ −1.00000 −0.171499
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 6.00000 0.986394 0.493197 0.869918i $$-0.335828\pi$$
0.493197 + 0.869918i $$0.335828\pi$$
$$38$$ 4.00000 0.648886
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 3.00000 0.468521 0.234261 0.972174i $$-0.424733\pi$$
0.234261 + 0.972174i $$0.424733\pi$$
$$42$$ 0 0
$$43$$ 1.00000 0.152499 0.0762493 0.997089i $$-0.475706\pi$$
0.0762493 + 0.997089i $$0.475706\pi$$
$$44$$ −2.00000 −0.301511
$$45$$ 0 0
$$46$$ 7.00000 1.03209
$$47$$ −12.0000 −1.75038 −0.875190 0.483779i $$-0.839264\pi$$
−0.875190 + 0.483779i $$0.839264\pi$$
$$48$$ 0 0
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ 0 0
$$52$$ −1.00000 −0.138675
$$53$$ 11.0000 1.51097 0.755483 0.655168i $$-0.227402\pi$$
0.755483 + 0.655168i $$0.227402\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 1.00000 0.133631
$$57$$ 0 0
$$58$$ −1.00000 −0.131306
$$59$$ 3.00000 0.390567 0.195283 0.980747i $$-0.437437\pi$$
0.195283 + 0.980747i $$0.437437\pi$$
$$60$$ 0 0
$$61$$ 5.00000 0.640184 0.320092 0.947386i $$-0.396286\pi$$
0.320092 + 0.947386i $$0.396286\pi$$
$$62$$ 3.00000 0.381000
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 12.0000 1.46603 0.733017 0.680211i $$-0.238112\pi$$
0.733017 + 0.680211i $$0.238112\pi$$
$$68$$ −1.00000 −0.121268
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −4.00000 −0.474713 −0.237356 0.971423i $$-0.576281\pi$$
−0.237356 + 0.971423i $$0.576281\pi$$
$$72$$ 0 0
$$73$$ 14.0000 1.63858 0.819288 0.573382i $$-0.194369\pi$$
0.819288 + 0.573382i $$0.194369\pi$$
$$74$$ 6.00000 0.697486
$$75$$ 0 0
$$76$$ 4.00000 0.458831
$$77$$ −2.00000 −0.227921
$$78$$ 0 0
$$79$$ −2.00000 −0.225018 −0.112509 0.993651i $$-0.535889\pi$$
−0.112509 + 0.993651i $$0.535889\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 3.00000 0.331295
$$83$$ −3.00000 −0.329293 −0.164646 0.986353i $$-0.552648\pi$$
−0.164646 + 0.986353i $$0.552648\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 1.00000 0.107833
$$87$$ 0 0
$$88$$ −2.00000 −0.213201
$$89$$ −10.0000 −1.06000 −0.529999 0.847998i $$-0.677808\pi$$
−0.529999 + 0.847998i $$0.677808\pi$$
$$90$$ 0 0
$$91$$ −1.00000 −0.104828
$$92$$ 7.00000 0.729800
$$93$$ 0 0
$$94$$ −12.0000 −1.23771
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 10.0000 1.01535 0.507673 0.861550i $$-0.330506\pi$$
0.507673 + 0.861550i $$0.330506\pi$$
$$98$$ 1.00000 0.101015
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$102$$ 0 0
$$103$$ 17.0000 1.67506 0.837530 0.546392i $$-0.183999\pi$$
0.837530 + 0.546392i $$0.183999\pi$$
$$104$$ −1.00000 −0.0980581
$$105$$ 0 0
$$106$$ 11.0000 1.06841
$$107$$ −18.0000 −1.74013 −0.870063 0.492941i $$-0.835922\pi$$
−0.870063 + 0.492941i $$0.835922\pi$$
$$108$$ 0 0
$$109$$ −4.00000 −0.383131 −0.191565 0.981480i $$-0.561356\pi$$
−0.191565 + 0.981480i $$0.561356\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 1.00000 0.0944911
$$113$$ 6.00000 0.564433 0.282216 0.959351i $$-0.408930\pi$$
0.282216 + 0.959351i $$0.408930\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −1.00000 −0.0928477
$$117$$ 0 0
$$118$$ 3.00000 0.276172
$$119$$ −1.00000 −0.0916698
$$120$$ 0 0
$$121$$ −7.00000 −0.636364
$$122$$ 5.00000 0.452679
$$123$$ 0 0
$$124$$ 3.00000 0.269408
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −14.0000 −1.24230 −0.621150 0.783692i $$-0.713334\pi$$
−0.621150 + 0.783692i $$0.713334\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −8.00000 −0.698963 −0.349482 0.936943i $$-0.613642\pi$$
−0.349482 + 0.936943i $$0.613642\pi$$
$$132$$ 0 0
$$133$$ 4.00000 0.346844
$$134$$ 12.0000 1.03664
$$135$$ 0 0
$$136$$ −1.00000 −0.0857493
$$137$$ 4.00000 0.341743 0.170872 0.985293i $$-0.445342\pi$$
0.170872 + 0.985293i $$0.445342\pi$$
$$138$$ 0 0
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −4.00000 −0.335673
$$143$$ 2.00000 0.167248
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 14.0000 1.15865
$$147$$ 0 0
$$148$$ 6.00000 0.493197
$$149$$ −5.00000 −0.409616 −0.204808 0.978802i $$-0.565657\pi$$
−0.204808 + 0.978802i $$0.565657\pi$$
$$150$$ 0 0
$$151$$ 22.0000 1.79033 0.895167 0.445730i $$-0.147056\pi$$
0.895167 + 0.445730i $$0.147056\pi$$
$$152$$ 4.00000 0.324443
$$153$$ 0 0
$$154$$ −2.00000 −0.161165
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −18.0000 −1.43656 −0.718278 0.695756i $$-0.755069\pi$$
−0.718278 + 0.695756i $$0.755069\pi$$
$$158$$ −2.00000 −0.159111
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 7.00000 0.551677
$$162$$ 0 0
$$163$$ 19.0000 1.48819 0.744097 0.668071i $$-0.232880\pi$$
0.744097 + 0.668071i $$0.232880\pi$$
$$164$$ 3.00000 0.234261
$$165$$ 0 0
$$166$$ −3.00000 −0.232845
$$167$$ −2.00000 −0.154765 −0.0773823 0.997001i $$-0.524656\pi$$
−0.0773823 + 0.997001i $$0.524656\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 1.00000 0.0762493
$$173$$ −12.0000 −0.912343 −0.456172 0.889892i $$-0.650780\pi$$
−0.456172 + 0.889892i $$0.650780\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −2.00000 −0.150756
$$177$$ 0 0
$$178$$ −10.0000 −0.749532
$$179$$ −10.0000 −0.747435 −0.373718 0.927543i $$-0.621917\pi$$
−0.373718 + 0.927543i $$0.621917\pi$$
$$180$$ 0 0
$$181$$ −18.0000 −1.33793 −0.668965 0.743294i $$-0.733262\pi$$
−0.668965 + 0.743294i $$0.733262\pi$$
$$182$$ −1.00000 −0.0741249
$$183$$ 0 0
$$184$$ 7.00000 0.516047
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 2.00000 0.146254
$$188$$ −12.0000 −0.875190
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 13.0000 0.940647 0.470323 0.882494i $$-0.344137\pi$$
0.470323 + 0.882494i $$0.344137\pi$$
$$192$$ 0 0
$$193$$ −2.00000 −0.143963 −0.0719816 0.997406i $$-0.522932\pi$$
−0.0719816 + 0.997406i $$0.522932\pi$$
$$194$$ 10.0000 0.717958
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ 27.0000 1.92367 0.961835 0.273629i $$-0.0882242\pi$$
0.961835 + 0.273629i $$0.0882242\pi$$
$$198$$ 0 0
$$199$$ −24.0000 −1.70131 −0.850657 0.525720i $$-0.823796\pi$$
−0.850657 + 0.525720i $$0.823796\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 0 0
$$203$$ −1.00000 −0.0701862
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 17.0000 1.18445
$$207$$ 0 0
$$208$$ −1.00000 −0.0693375
$$209$$ −8.00000 −0.553372
$$210$$ 0 0
$$211$$ 15.0000 1.03264 0.516321 0.856395i $$-0.327301\pi$$
0.516321 + 0.856395i $$0.327301\pi$$
$$212$$ 11.0000 0.755483
$$213$$ 0 0
$$214$$ −18.0000 −1.23045
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 3.00000 0.203653
$$218$$ −4.00000 −0.270914
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 1.00000 0.0672673
$$222$$ 0 0
$$223$$ 13.0000 0.870544 0.435272 0.900299i $$-0.356652\pi$$
0.435272 + 0.900299i $$0.356652\pi$$
$$224$$ 1.00000 0.0668153
$$225$$ 0 0
$$226$$ 6.00000 0.399114
$$227$$ 7.00000 0.464606 0.232303 0.972643i $$-0.425374\pi$$
0.232303 + 0.972643i $$0.425374\pi$$
$$228$$ 0 0
$$229$$ 14.0000 0.925146 0.462573 0.886581i $$-0.346926\pi$$
0.462573 + 0.886581i $$0.346926\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −1.00000 −0.0656532
$$233$$ 20.0000 1.31024 0.655122 0.755523i $$-0.272617\pi$$
0.655122 + 0.755523i $$0.272617\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 3.00000 0.195283
$$237$$ 0 0
$$238$$ −1.00000 −0.0648204
$$239$$ 24.0000 1.55243 0.776215 0.630468i $$-0.217137\pi$$
0.776215 + 0.630468i $$0.217137\pi$$
$$240$$ 0 0
$$241$$ −26.0000 −1.67481 −0.837404 0.546585i $$-0.815928\pi$$
−0.837404 + 0.546585i $$0.815928\pi$$
$$242$$ −7.00000 −0.449977
$$243$$ 0 0
$$244$$ 5.00000 0.320092
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −4.00000 −0.254514
$$248$$ 3.00000 0.190500
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 15.0000 0.946792 0.473396 0.880850i $$-0.343028\pi$$
0.473396 + 0.880850i $$0.343028\pi$$
$$252$$ 0 0
$$253$$ −14.0000 −0.880172
$$254$$ −14.0000 −0.878438
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ −5.00000 −0.311891 −0.155946 0.987766i $$-0.549842\pi$$
−0.155946 + 0.987766i $$0.549842\pi$$
$$258$$ 0 0
$$259$$ 6.00000 0.372822
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −8.00000 −0.494242
$$263$$ 11.0000 0.678289 0.339145 0.940734i $$-0.389862\pi$$
0.339145 + 0.940734i $$0.389862\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 4.00000 0.245256
$$267$$ 0 0
$$268$$ 12.0000 0.733017
$$269$$ −18.0000 −1.09748 −0.548740 0.835993i $$-0.684892\pi$$
−0.548740 + 0.835993i $$0.684892\pi$$
$$270$$ 0 0
$$271$$ −20.0000 −1.21491 −0.607457 0.794353i $$-0.707810\pi$$
−0.607457 + 0.794353i $$0.707810\pi$$
$$272$$ −1.00000 −0.0606339
$$273$$ 0 0
$$274$$ 4.00000 0.241649
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 2.00000 0.120168 0.0600842 0.998193i $$-0.480863\pi$$
0.0600842 + 0.998193i $$0.480863\pi$$
$$278$$ −4.00000 −0.239904
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$282$$ 0 0
$$283$$ 20.0000 1.18888 0.594438 0.804141i $$-0.297374\pi$$
0.594438 + 0.804141i $$0.297374\pi$$
$$284$$ −4.00000 −0.237356
$$285$$ 0 0
$$286$$ 2.00000 0.118262
$$287$$ 3.00000 0.177084
$$288$$ 0 0
$$289$$ −16.0000 −0.941176
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 14.0000 0.819288
$$293$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 6.00000 0.348743
$$297$$ 0 0
$$298$$ −5.00000 −0.289642
$$299$$ −7.00000 −0.404820
$$300$$ 0 0
$$301$$ 1.00000 0.0576390
$$302$$ 22.0000 1.26596
$$303$$ 0 0
$$304$$ 4.00000 0.229416
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 22.0000 1.25561 0.627803 0.778372i $$-0.283954\pi$$
0.627803 + 0.778372i $$0.283954\pi$$
$$308$$ −2.00000 −0.113961
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −34.0000 −1.92796 −0.963982 0.265969i $$-0.914308\pi$$
−0.963982 + 0.265969i $$0.914308\pi$$
$$312$$ 0 0
$$313$$ −18.0000 −1.01742 −0.508710 0.860938i $$-0.669877\pi$$
−0.508710 + 0.860938i $$0.669877\pi$$
$$314$$ −18.0000 −1.01580
$$315$$ 0 0
$$316$$ −2.00000 −0.112509
$$317$$ −17.0000 −0.954815 −0.477408 0.878682i $$-0.658423\pi$$
−0.477408 + 0.878682i $$0.658423\pi$$
$$318$$ 0 0
$$319$$ 2.00000 0.111979
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 7.00000 0.390095
$$323$$ −4.00000 −0.222566
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 19.0000 1.05231
$$327$$ 0 0
$$328$$ 3.00000 0.165647
$$329$$ −12.0000 −0.661581
$$330$$ 0 0
$$331$$ −3.00000 −0.164895 −0.0824475 0.996595i $$-0.526274\pi$$
−0.0824475 + 0.996595i $$0.526274\pi$$
$$332$$ −3.00000 −0.164646
$$333$$ 0 0
$$334$$ −2.00000 −0.109435
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −27.0000 −1.47078 −0.735392 0.677642i $$-0.763002\pi$$
−0.735392 + 0.677642i $$0.763002\pi$$
$$338$$ −12.0000 −0.652714
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −6.00000 −0.324918
$$342$$ 0 0
$$343$$ 1.00000 0.0539949
$$344$$ 1.00000 0.0539164
$$345$$ 0 0
$$346$$ −12.0000 −0.645124
$$347$$ 26.0000 1.39575 0.697877 0.716218i $$-0.254128\pi$$
0.697877 + 0.716218i $$0.254128\pi$$
$$348$$ 0 0
$$349$$ −1.00000 −0.0535288 −0.0267644 0.999642i $$-0.508520\pi$$
−0.0267644 + 0.999642i $$0.508520\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −2.00000 −0.106600
$$353$$ 10.0000 0.532246 0.266123 0.963939i $$-0.414257\pi$$
0.266123 + 0.963939i $$0.414257\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −10.0000 −0.529999
$$357$$ 0 0
$$358$$ −10.0000 −0.528516
$$359$$ 9.00000 0.475002 0.237501 0.971387i $$-0.423672\pi$$
0.237501 + 0.971387i $$0.423672\pi$$
$$360$$ 0 0
$$361$$ −3.00000 −0.157895
$$362$$ −18.0000 −0.946059
$$363$$ 0 0
$$364$$ −1.00000 −0.0524142
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −13.0000 −0.678594 −0.339297 0.940679i $$-0.610189\pi$$
−0.339297 + 0.940679i $$0.610189\pi$$
$$368$$ 7.00000 0.364900
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 11.0000 0.571092
$$372$$ 0 0
$$373$$ 4.00000 0.207112 0.103556 0.994624i $$-0.466978\pi$$
0.103556 + 0.994624i $$0.466978\pi$$
$$374$$ 2.00000 0.103418
$$375$$ 0 0
$$376$$ −12.0000 −0.618853
$$377$$ 1.00000 0.0515026
$$378$$ 0 0
$$379$$ −5.00000 −0.256833 −0.128416 0.991720i $$-0.540989\pi$$
−0.128416 + 0.991720i $$0.540989\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 13.0000 0.665138
$$383$$ −20.0000 −1.02195 −0.510976 0.859595i $$-0.670716\pi$$
−0.510976 + 0.859595i $$0.670716\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −2.00000 −0.101797
$$387$$ 0 0
$$388$$ 10.0000 0.507673
$$389$$ −22.0000 −1.11544 −0.557722 0.830028i $$-0.688325\pi$$
−0.557722 + 0.830028i $$0.688325\pi$$
$$390$$ 0 0
$$391$$ −7.00000 −0.354005
$$392$$ 1.00000 0.0505076
$$393$$ 0 0
$$394$$ 27.0000 1.36024
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −3.00000 −0.150566 −0.0752828 0.997162i $$-0.523986\pi$$
−0.0752828 + 0.997162i $$0.523986\pi$$
$$398$$ −24.0000 −1.20301
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −16.0000 −0.799002 −0.399501 0.916733i $$-0.630817\pi$$
−0.399501 + 0.916733i $$0.630817\pi$$
$$402$$ 0 0
$$403$$ −3.00000 −0.149441
$$404$$ 0 0
$$405$$ 0 0
$$406$$ −1.00000 −0.0496292
$$407$$ −12.0000 −0.594818
$$408$$ 0 0
$$409$$ −14.0000 −0.692255 −0.346128 0.938187i $$-0.612504\pi$$
−0.346128 + 0.938187i $$0.612504\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 17.0000 0.837530
$$413$$ 3.00000 0.147620
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −1.00000 −0.0490290
$$417$$ 0 0
$$418$$ −8.00000 −0.391293
$$419$$ 25.0000 1.22133 0.610665 0.791889i $$-0.290902\pi$$
0.610665 + 0.791889i $$0.290902\pi$$
$$420$$ 0 0
$$421$$ 34.0000 1.65706 0.828529 0.559946i $$-0.189178\pi$$
0.828529 + 0.559946i $$0.189178\pi$$
$$422$$ 15.0000 0.730189
$$423$$ 0 0
$$424$$ 11.0000 0.534207
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 5.00000 0.241967
$$428$$ −18.0000 −0.870063
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −27.0000 −1.30054 −0.650272 0.759701i $$-0.725345\pi$$
−0.650272 + 0.759701i $$0.725345\pi$$
$$432$$ 0 0
$$433$$ −34.0000 −1.63394 −0.816968 0.576683i $$-0.804347\pi$$
−0.816968 + 0.576683i $$0.804347\pi$$
$$434$$ 3.00000 0.144005
$$435$$ 0 0
$$436$$ −4.00000 −0.191565
$$437$$ 28.0000 1.33942
$$438$$ 0 0
$$439$$ 7.00000 0.334092 0.167046 0.985949i $$-0.446577\pi$$
0.167046 + 0.985949i $$0.446577\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 1.00000 0.0475651
$$443$$ −12.0000 −0.570137 −0.285069 0.958507i $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 13.0000 0.615568
$$447$$ 0 0
$$448$$ 1.00000 0.0472456
$$449$$ 16.0000 0.755087 0.377543 0.925992i $$-0.376769\pi$$
0.377543 + 0.925992i $$0.376769\pi$$
$$450$$ 0 0
$$451$$ −6.00000 −0.282529
$$452$$ 6.00000 0.282216
$$453$$ 0 0
$$454$$ 7.00000 0.328526
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 17.0000 0.795226 0.397613 0.917553i $$-0.369839\pi$$
0.397613 + 0.917553i $$0.369839\pi$$
$$458$$ 14.0000 0.654177
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −32.0000 −1.49039 −0.745194 0.666847i $$-0.767643\pi$$
−0.745194 + 0.666847i $$0.767643\pi$$
$$462$$ 0 0
$$463$$ −36.0000 −1.67306 −0.836531 0.547920i $$-0.815420\pi$$
−0.836531 + 0.547920i $$0.815420\pi$$
$$464$$ −1.00000 −0.0464238
$$465$$ 0 0
$$466$$ 20.0000 0.926482
$$467$$ −39.0000 −1.80470 −0.902352 0.430999i $$-0.858161\pi$$
−0.902352 + 0.430999i $$0.858161\pi$$
$$468$$ 0 0
$$469$$ 12.0000 0.554109
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 3.00000 0.138086
$$473$$ −2.00000 −0.0919601
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −1.00000 −0.0458349
$$477$$ 0 0
$$478$$ 24.0000 1.09773
$$479$$ −8.00000 −0.365529 −0.182765 0.983157i $$-0.558505\pi$$
−0.182765 + 0.983157i $$0.558505\pi$$
$$480$$ 0 0
$$481$$ −6.00000 −0.273576
$$482$$ −26.0000 −1.18427
$$483$$ 0 0
$$484$$ −7.00000 −0.318182
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −22.0000 −0.996915 −0.498458 0.866914i $$-0.666100\pi$$
−0.498458 + 0.866914i $$0.666100\pi$$
$$488$$ 5.00000 0.226339
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −30.0000 −1.35388 −0.676941 0.736038i $$-0.736695\pi$$
−0.676941 + 0.736038i $$0.736695\pi$$
$$492$$ 0 0
$$493$$ 1.00000 0.0450377
$$494$$ −4.00000 −0.179969
$$495$$ 0 0
$$496$$ 3.00000 0.134704
$$497$$ −4.00000 −0.179425
$$498$$ 0 0
$$499$$ −27.0000 −1.20869 −0.604343 0.796724i $$-0.706564\pi$$
−0.604343 + 0.796724i $$0.706564\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 15.0000 0.669483
$$503$$ −2.00000 −0.0891756 −0.0445878 0.999005i $$-0.514197\pi$$
−0.0445878 + 0.999005i $$0.514197\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −14.0000 −0.622376
$$507$$ 0 0
$$508$$ −14.0000 −0.621150
$$509$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$510$$ 0 0
$$511$$ 14.0000 0.619324
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ −5.00000 −0.220541
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 24.0000 1.05552
$$518$$ 6.00000 0.263625
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −39.0000 −1.70862 −0.854311 0.519763i $$-0.826020\pi$$
−0.854311 + 0.519763i $$0.826020\pi$$
$$522$$ 0 0
$$523$$ 8.00000 0.349816 0.174908 0.984585i $$-0.444037\pi$$
0.174908 + 0.984585i $$0.444037\pi$$
$$524$$ −8.00000 −0.349482
$$525$$ 0 0
$$526$$ 11.0000 0.479623
$$527$$ −3.00000 −0.130682
$$528$$ 0 0
$$529$$ 26.0000 1.13043
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 4.00000 0.173422
$$533$$ −3.00000 −0.129944
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 12.0000 0.518321
$$537$$ 0 0
$$538$$ −18.0000 −0.776035
$$539$$ −2.00000 −0.0861461
$$540$$ 0 0
$$541$$ −28.0000 −1.20381 −0.601907 0.798566i $$-0.705592\pi$$
−0.601907 + 0.798566i $$0.705592\pi$$
$$542$$ −20.0000 −0.859074
$$543$$ 0 0
$$544$$ −1.00000 −0.0428746
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 15.0000 0.641354 0.320677 0.947189i $$-0.396090\pi$$
0.320677 + 0.947189i $$0.396090\pi$$
$$548$$ 4.00000 0.170872
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −4.00000 −0.170406
$$552$$ 0 0
$$553$$ −2.00000 −0.0850487
$$554$$ 2.00000 0.0849719
$$555$$ 0 0
$$556$$ −4.00000 −0.169638
$$557$$ 18.0000 0.762684 0.381342 0.924434i $$-0.375462\pi$$
0.381342 + 0.924434i $$0.375462\pi$$
$$558$$ 0 0
$$559$$ −1.00000 −0.0422955
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 0 0
$$563$$ 41.0000 1.72794 0.863972 0.503540i $$-0.167969\pi$$
0.863972 + 0.503540i $$0.167969\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 20.0000 0.840663
$$567$$ 0 0
$$568$$ −4.00000 −0.167836
$$569$$ −14.0000 −0.586911 −0.293455 0.955973i $$-0.594805\pi$$
−0.293455 + 0.955973i $$0.594805\pi$$
$$570$$ 0 0
$$571$$ 19.0000 0.795125 0.397563 0.917575i $$-0.369856\pi$$
0.397563 + 0.917575i $$0.369856\pi$$
$$572$$ 2.00000 0.0836242
$$573$$ 0 0
$$574$$ 3.00000 0.125218
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −34.0000 −1.41544 −0.707719 0.706494i $$-0.750276\pi$$
−0.707719 + 0.706494i $$0.750276\pi$$
$$578$$ −16.0000 −0.665512
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −3.00000 −0.124461
$$582$$ 0 0
$$583$$ −22.0000 −0.911147
$$584$$ 14.0000 0.579324
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −27.0000 −1.11441 −0.557205 0.830375i $$-0.688126\pi$$
−0.557205 + 0.830375i $$0.688126\pi$$
$$588$$ 0 0
$$589$$ 12.0000 0.494451
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 6.00000 0.246598
$$593$$ −18.0000 −0.739171 −0.369586 0.929197i $$-0.620500\pi$$
−0.369586 + 0.929197i $$0.620500\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −5.00000 −0.204808
$$597$$ 0 0
$$598$$ −7.00000 −0.286251
$$599$$ 45.0000 1.83865 0.919325 0.393499i $$-0.128735\pi$$
0.919325 + 0.393499i $$0.128735\pi$$
$$600$$ 0 0
$$601$$ 28.0000 1.14214 0.571072 0.820900i $$-0.306528\pi$$
0.571072 + 0.820900i $$0.306528\pi$$
$$602$$ 1.00000 0.0407570
$$603$$ 0 0
$$604$$ 22.0000 0.895167
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −28.0000 −1.13648 −0.568242 0.822861i $$-0.692376\pi$$
−0.568242 + 0.822861i $$0.692376\pi$$
$$608$$ 4.00000 0.162221
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 12.0000 0.485468
$$612$$ 0 0
$$613$$ −18.0000 −0.727013 −0.363507 0.931592i $$-0.618421\pi$$
−0.363507 + 0.931592i $$0.618421\pi$$
$$614$$ 22.0000 0.887848
$$615$$ 0 0
$$616$$ −2.00000 −0.0805823
$$617$$ 14.0000 0.563619 0.281809 0.959470i $$-0.409065\pi$$
0.281809 + 0.959470i $$0.409065\pi$$
$$618$$ 0 0
$$619$$ 38.0000 1.52735 0.763674 0.645601i $$-0.223393\pi$$
0.763674 + 0.645601i $$0.223393\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −34.0000 −1.36328
$$623$$ −10.0000 −0.400642
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −18.0000 −0.719425
$$627$$ 0 0
$$628$$ −18.0000 −0.718278
$$629$$ −6.00000 −0.239236
$$630$$ 0 0
$$631$$ 26.0000 1.03504 0.517522 0.855670i $$-0.326855\pi$$
0.517522 + 0.855670i $$0.326855\pi$$
$$632$$ −2.00000 −0.0795557
$$633$$ 0 0
$$634$$ −17.0000 −0.675156
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −1.00000 −0.0396214
$$638$$ 2.00000 0.0791808
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 26.0000 1.02694 0.513469 0.858108i $$-0.328360\pi$$
0.513469 + 0.858108i $$0.328360\pi$$
$$642$$ 0 0
$$643$$ −2.00000 −0.0788723 −0.0394362 0.999222i $$-0.512556\pi$$
−0.0394362 + 0.999222i $$0.512556\pi$$
$$644$$ 7.00000 0.275839
$$645$$ 0 0
$$646$$ −4.00000 −0.157378
$$647$$ 6.00000 0.235884 0.117942 0.993020i $$-0.462370\pi$$
0.117942 + 0.993020i $$0.462370\pi$$
$$648$$ 0 0
$$649$$ −6.00000 −0.235521
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 19.0000 0.744097
$$653$$ 14.0000 0.547862 0.273931 0.961749i $$-0.411676\pi$$
0.273931 + 0.961749i $$0.411676\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 3.00000 0.117130
$$657$$ 0 0
$$658$$ −12.0000 −0.467809
$$659$$ 6.00000 0.233727 0.116863 0.993148i $$-0.462716\pi$$
0.116863 + 0.993148i $$0.462716\pi$$
$$660$$ 0 0
$$661$$ −22.0000 −0.855701 −0.427850 0.903850i $$-0.640729\pi$$
−0.427850 + 0.903850i $$0.640729\pi$$
$$662$$ −3.00000 −0.116598
$$663$$ 0 0
$$664$$ −3.00000 −0.116423
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −7.00000 −0.271041
$$668$$ −2.00000 −0.0773823
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −10.0000 −0.386046
$$672$$ 0 0
$$673$$ −27.0000 −1.04077 −0.520387 0.853931i $$-0.674212\pi$$
−0.520387 + 0.853931i $$0.674212\pi$$
$$674$$ −27.0000 −1.04000
$$675$$ 0 0
$$676$$ −12.0000 −0.461538
$$677$$ −20.0000 −0.768662 −0.384331 0.923195i $$-0.625568\pi$$
−0.384331 + 0.923195i $$0.625568\pi$$
$$678$$ 0 0
$$679$$ 10.0000 0.383765
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −6.00000 −0.229752
$$683$$ −6.00000 −0.229584 −0.114792 0.993390i $$-0.536620\pi$$
−0.114792 + 0.993390i $$0.536620\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 1.00000 0.0381802
$$687$$ 0 0
$$688$$ 1.00000 0.0381246
$$689$$ −11.0000 −0.419067
$$690$$ 0 0
$$691$$ −42.0000 −1.59776 −0.798878 0.601494i $$-0.794573\pi$$
−0.798878 + 0.601494i $$0.794573\pi$$
$$692$$ −12.0000 −0.456172
$$693$$ 0 0
$$694$$ 26.0000 0.986947
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −3.00000 −0.113633
$$698$$ −1.00000 −0.0378506
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 39.0000 1.47301 0.736505 0.676432i $$-0.236475\pi$$
0.736505 + 0.676432i $$0.236475\pi$$
$$702$$ 0 0
$$703$$ 24.0000 0.905177
$$704$$ −2.00000 −0.0753778
$$705$$ 0 0
$$706$$ 10.0000 0.376355
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 4.00000 0.150223 0.0751116 0.997175i $$-0.476069\pi$$
0.0751116 + 0.997175i $$0.476069\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ −10.0000 −0.374766
$$713$$ 21.0000 0.786456
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −10.0000 −0.373718
$$717$$ 0 0
$$718$$ 9.00000 0.335877
$$719$$ 34.0000 1.26799 0.633993 0.773339i $$-0.281415\pi$$
0.633993 + 0.773339i $$0.281415\pi$$
$$720$$ 0 0
$$721$$ 17.0000 0.633113
$$722$$ −3.00000 −0.111648
$$723$$ 0 0
$$724$$ −18.0000 −0.668965
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 1.00000 0.0370879 0.0185440 0.999828i $$-0.494097\pi$$
0.0185440 + 0.999828i $$0.494097\pi$$
$$728$$ −1.00000 −0.0370625
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −1.00000 −0.0369863
$$732$$ 0 0
$$733$$ −19.0000 −0.701781 −0.350891 0.936416i $$-0.614121\pi$$
−0.350891 + 0.936416i $$0.614121\pi$$
$$734$$ −13.0000 −0.479839
$$735$$ 0 0
$$736$$ 7.00000 0.258023
$$737$$ −24.0000 −0.884051
$$738$$ 0 0
$$739$$ 11.0000 0.404642 0.202321 0.979319i $$-0.435152\pi$$
0.202321 + 0.979319i $$0.435152\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 11.0000 0.403823
$$743$$ −3.00000 −0.110059 −0.0550297 0.998485i $$-0.517525\pi$$
−0.0550297 + 0.998485i $$0.517525\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 4.00000 0.146450
$$747$$ 0 0
$$748$$ 2.00000 0.0731272
$$749$$ −18.0000 −0.657706
$$750$$ 0 0
$$751$$ −20.0000 −0.729810 −0.364905 0.931045i $$-0.618899\pi$$
−0.364905 + 0.931045i $$0.618899\pi$$
$$752$$ −12.0000 −0.437595
$$753$$ 0 0
$$754$$ 1.00000 0.0364179
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 2.00000 0.0726912 0.0363456 0.999339i $$-0.488428\pi$$
0.0363456 + 0.999339i $$0.488428\pi$$
$$758$$ −5.00000 −0.181608
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −34.0000 −1.23250 −0.616250 0.787551i $$-0.711349\pi$$
−0.616250 + 0.787551i $$0.711349\pi$$
$$762$$ 0 0
$$763$$ −4.00000 −0.144810
$$764$$ 13.0000 0.470323
$$765$$ 0 0
$$766$$ −20.0000 −0.722629
$$767$$ −3.00000 −0.108324
$$768$$ 0 0
$$769$$ 40.0000 1.44244 0.721218 0.692708i $$-0.243582\pi$$
0.721218 + 0.692708i $$0.243582\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −2.00000 −0.0719816
$$773$$ −20.0000 −0.719350 −0.359675 0.933078i $$-0.617112\pi$$
−0.359675 + 0.933078i $$0.617112\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 10.0000 0.358979
$$777$$ 0 0
$$778$$ −22.0000 −0.788738
$$779$$ 12.0000 0.429945
$$780$$ 0 0
$$781$$ 8.00000 0.286263
$$782$$ −7.00000 −0.250319
$$783$$ 0 0
$$784$$ 1.00000 0.0357143
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −38.0000 −1.35455 −0.677277 0.735728i $$-0.736840\pi$$
−0.677277 + 0.735728i $$0.736840\pi$$
$$788$$ 27.0000 0.961835
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 6.00000 0.213335
$$792$$ 0 0
$$793$$ −5.00000 −0.177555
$$794$$ −3.00000 −0.106466
$$795$$ 0 0
$$796$$ −24.0000 −0.850657
$$797$$ 42.0000 1.48772 0.743858 0.668338i $$-0.232994\pi$$
0.743858 + 0.668338i $$0.232994\pi$$
$$798$$ 0 0
$$799$$ 12.0000 0.424529
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −16.0000 −0.564980
$$803$$ −28.0000 −0.988099
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −3.00000 −0.105670
$$807$$ 0 0
$$808$$ 0 0
$$809$$ −20.0000 −0.703163 −0.351581 0.936157i $$-0.614356\pi$$
−0.351581 + 0.936157i $$0.614356\pi$$
$$810$$ 0 0
$$811$$ −14.0000 −0.491606 −0.245803 0.969320i $$-0.579052\pi$$
−0.245803 + 0.969320i $$0.579052\pi$$
$$812$$ −1.00000 −0.0350931
$$813$$ 0 0
$$814$$ −12.0000 −0.420600
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 4.00000 0.139942
$$818$$ −14.0000 −0.489499
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 38.0000 1.32621 0.663105 0.748527i $$-0.269238\pi$$
0.663105 + 0.748527i $$0.269238\pi$$
$$822$$ 0 0
$$823$$ 18.0000 0.627441 0.313720 0.949515i $$-0.398425\pi$$
0.313720 + 0.949515i $$0.398425\pi$$
$$824$$ 17.0000 0.592223
$$825$$ 0 0
$$826$$ 3.00000 0.104383
$$827$$ −50.0000 −1.73867 −0.869335 0.494223i $$-0.835453\pi$$
−0.869335 + 0.494223i $$0.835453\pi$$
$$828$$ 0 0
$$829$$ 3.00000 0.104194 0.0520972 0.998642i $$-0.483409\pi$$
0.0520972 + 0.998642i $$0.483409\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −1.00000 −0.0346688
$$833$$ −1.00000 −0.0346479
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −8.00000 −0.276686
$$837$$ 0 0
$$838$$ 25.0000 0.863611
$$839$$ 40.0000 1.38095 0.690477 0.723355i $$-0.257401\pi$$
0.690477 + 0.723355i $$0.257401\pi$$
$$840$$ 0 0
$$841$$ −28.0000 −0.965517
$$842$$ 34.0000 1.17172
$$843$$ 0 0
$$844$$ 15.0000 0.516321
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −7.00000 −0.240523
$$848$$ 11.0000 0.377742
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 42.0000 1.43974
$$852$$ 0 0
$$853$$ −43.0000 −1.47229 −0.736146 0.676823i $$-0.763356\pi$$
−0.736146 + 0.676823i $$0.763356\pi$$
$$854$$ 5.00000 0.171096
$$855$$ 0 0
$$856$$ −18.0000 −0.615227
$$857$$ −46.0000 −1.57133 −0.785665 0.618652i $$-0.787679\pi$$
−0.785665 + 0.618652i $$0.787679\pi$$
$$858$$ 0 0
$$859$$ 40.0000 1.36478 0.682391 0.730987i $$-0.260940\pi$$
0.682391 + 0.730987i $$0.260940\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ −27.0000 −0.919624
$$863$$ 32.0000 1.08929 0.544646 0.838666i $$-0.316664\pi$$
0.544646 + 0.838666i $$0.316664\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −34.0000 −1.15537
$$867$$ 0 0
$$868$$ 3.00000 0.101827
$$869$$ 4.00000 0.135691
$$870$$ 0 0
$$871$$ −12.0000 −0.406604
$$872$$ −4.00000 −0.135457
$$873$$ 0 0
$$874$$ 28.0000 0.947114
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −32.0000 −1.08056 −0.540282 0.841484i $$-0.681682\pi$$
−0.540282 + 0.841484i $$0.681682\pi$$
$$878$$ 7.00000 0.236239
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −5.00000 −0.168454 −0.0842271 0.996447i $$-0.526842\pi$$
−0.0842271 + 0.996447i $$0.526842\pi$$
$$882$$ 0 0
$$883$$ −29.0000 −0.975928 −0.487964 0.872864i $$-0.662260\pi$$
−0.487964 + 0.872864i $$0.662260\pi$$
$$884$$ 1.00000 0.0336336
$$885$$ 0 0
$$886$$ −12.0000 −0.403148
$$887$$ 50.0000 1.67884 0.839418 0.543487i $$-0.182896\pi$$
0.839418 + 0.543487i $$0.182896\pi$$
$$888$$ 0 0
$$889$$ −14.0000 −0.469545
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 13.0000 0.435272
$$893$$ −48.0000 −1.60626
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 1.00000 0.0334077
$$897$$ 0 0
$$898$$ 16.0000 0.533927
$$899$$ −3.00000 −0.100056
$$900$$ 0 0
$$901$$ −11.0000 −0.366463
$$902$$ −6.00000 −0.199778
$$903$$ 0 0
$$904$$ 6.00000 0.199557
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 11.0000 0.365249 0.182625 0.983183i $$-0.441541\pi$$
0.182625 + 0.983183i $$0.441541\pi$$
$$908$$ 7.00000 0.232303
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 19.0000 0.629498 0.314749 0.949175i $$-0.398080\pi$$
0.314749 + 0.949175i $$0.398080\pi$$
$$912$$ 0 0
$$913$$ 6.00000 0.198571
$$914$$ 17.0000 0.562310
$$915$$ 0 0
$$916$$ 14.0000 0.462573
$$917$$ −8.00000 −0.264183
$$918$$ 0 0
$$919$$ −20.0000 −0.659739 −0.329870 0.944027i $$-0.607005\pi$$
−0.329870 + 0.944027i $$0.607005\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ −32.0000 −1.05386
$$923$$ 4.00000 0.131662
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −36.0000 −1.18303
$$927$$ 0 0
$$928$$ −1.00000 −0.0328266
$$929$$ −25.0000 −0.820223 −0.410112 0.912035i $$-0.634510\pi$$
−0.410112 + 0.912035i $$0.634510\pi$$
$$930$$ 0 0
$$931$$ 4.00000 0.131095
$$932$$ 20.0000 0.655122
$$933$$ 0 0
$$934$$ −39.0000 −1.27612
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 14.0000 0.457360 0.228680 0.973502i $$-0.426559\pi$$
0.228680 + 0.973502i $$0.426559\pi$$
$$938$$ 12.0000 0.391814
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$942$$ 0 0
$$943$$ 21.0000 0.683854
$$944$$ 3.00000 0.0976417
$$945$$ 0 0
$$946$$ −2.00000 −0.0650256
$$947$$ −32.0000 −1.03986 −0.519930 0.854209i $$-0.674042\pi$$
−0.519930 + 0.854209i $$0.674042\pi$$
$$948$$ 0 0
$$949$$ −14.0000 −0.454459
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −1.00000 −0.0324102
$$953$$ −40.0000 −1.29573 −0.647864 0.761756i $$-0.724337\pi$$
−0.647864 + 0.761756i $$0.724337\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 24.0000 0.776215
$$957$$ 0 0
$$958$$ −8.00000 −0.258468
$$959$$ 4.00000 0.129167
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ −6.00000 −0.193448
$$963$$ 0 0
$$964$$ −26.0000 −0.837404
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 44.0000 1.41494 0.707472 0.706741i $$-0.249835\pi$$
0.707472 + 0.706741i $$0.249835\pi$$
$$968$$ −7.00000 −0.224989
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 44.0000 1.41203 0.706014 0.708198i $$-0.250492\pi$$
0.706014 + 0.708198i $$0.250492\pi$$
$$972$$ 0 0
$$973$$ −4.00000 −0.128234
$$974$$ −22.0000 −0.704925
$$975$$ 0 0
$$976$$ 5.00000 0.160046
$$977$$ 42.0000 1.34370 0.671850 0.740688i $$-0.265500\pi$$
0.671850 + 0.740688i $$0.265500\pi$$
$$978$$ 0 0
$$979$$ 20.0000 0.639203
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −30.0000 −0.957338
$$983$$ 24.0000 0.765481 0.382741 0.923856i $$-0.374980\pi$$
0.382741 + 0.923856i $$0.374980\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 1.00000 0.0318465
$$987$$ 0 0
$$988$$ −4.00000 −0.127257
$$989$$ 7.00000 0.222587
$$990$$ 0 0
$$991$$ 4.00000 0.127064 0.0635321 0.997980i $$-0.479763\pi$$
0.0635321 + 0.997980i $$0.479763\pi$$
$$992$$ 3.00000 0.0952501
$$993$$ 0 0
$$994$$ −4.00000 −0.126872
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 2.00000 0.0633406 0.0316703 0.999498i $$-0.489917\pi$$
0.0316703 + 0.999498i $$0.489917\pi$$
$$998$$ −27.0000 −0.854670
$$999$$ 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 3150.2.a.bl.1.1 1
3.2 odd 2 1050.2.a.e.1.1 1
5.2 odd 4 3150.2.g.g.2899.2 2
5.3 odd 4 3150.2.g.g.2899.1 2
5.4 even 2 3150.2.a.c.1.1 1
12.11 even 2 8400.2.a.bt.1.1 1
15.2 even 4 1050.2.g.j.799.1 2
15.8 even 4 1050.2.g.j.799.2 2
15.14 odd 2 1050.2.a.o.1.1 yes 1
21.20 even 2 7350.2.a.bj.1.1 1
60.59 even 2 8400.2.a.t.1.1 1
105.104 even 2 7350.2.a.ca.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
1050.2.a.e.1.1 1 3.2 odd 2
1050.2.a.o.1.1 yes 1 15.14 odd 2
1050.2.g.j.799.1 2 15.2 even 4
1050.2.g.j.799.2 2 15.8 even 4
3150.2.a.c.1.1 1 5.4 even 2
3150.2.a.bl.1.1 1 1.1 even 1 trivial
3150.2.g.g.2899.1 2 5.3 odd 4
3150.2.g.g.2899.2 2 5.2 odd 4
7350.2.a.bj.1.1 1 21.20 even 2
7350.2.a.ca.1.1 1 105.104 even 2
8400.2.a.t.1.1 1 60.59 even 2
8400.2.a.bt.1.1 1 12.11 even 2