# Properties

 Label 4410.2.a.bg.1.1 Level $4410$ Weight $2$ Character 4410.1 Self dual yes Analytic conductor $35.214$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$4410 = 2 \cdot 3^{2} \cdot 5 \cdot 7^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 4410.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$35.2140272914$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 70) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 4410.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{8} +O(q^{10})$$ $$q+1.00000 q^{2} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{8} +1.00000 q^{10} -3.00000 q^{11} +1.00000 q^{13} +1.00000 q^{16} -6.00000 q^{17} +1.00000 q^{19} +1.00000 q^{20} -3.00000 q^{22} -9.00000 q^{23} +1.00000 q^{25} +1.00000 q^{26} -6.00000 q^{29} -8.00000 q^{31} +1.00000 q^{32} -6.00000 q^{34} -7.00000 q^{37} +1.00000 q^{38} +1.00000 q^{40} +3.00000 q^{41} +2.00000 q^{43} -3.00000 q^{44} -9.00000 q^{46} +9.00000 q^{47} +1.00000 q^{50} +1.00000 q^{52} -9.00000 q^{53} -3.00000 q^{55} -6.00000 q^{58} -8.00000 q^{61} -8.00000 q^{62} +1.00000 q^{64} +1.00000 q^{65} +8.00000 q^{67} -6.00000 q^{68} +4.00000 q^{73} -7.00000 q^{74} +1.00000 q^{76} -10.0000 q^{79} +1.00000 q^{80} +3.00000 q^{82} -6.00000 q^{85} +2.00000 q^{86} -3.00000 q^{88} +6.00000 q^{89} -9.00000 q^{92} +9.00000 q^{94} +1.00000 q^{95} +10.0000 q^{97} +O(q^{100})$$

## 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
$$5$$ 1.00000 0.447214
$$6$$ 0 0
$$7$$ 0 0
$$8$$ 1.00000 0.353553
$$9$$ 0 0
$$10$$ 1.00000 0.316228
$$11$$ −3.00000 −0.904534 −0.452267 0.891883i $$-0.649385\pi$$
−0.452267 + 0.891883i $$0.649385\pi$$
$$12$$ 0 0
$$13$$ 1.00000 0.277350 0.138675 0.990338i $$-0.455716\pi$$
0.138675 + 0.990338i $$0.455716\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −6.00000 −1.45521 −0.727607 0.685994i $$-0.759367\pi$$
−0.727607 + 0.685994i $$0.759367\pi$$
$$18$$ 0 0
$$19$$ 1.00000 0.229416 0.114708 0.993399i $$-0.463407\pi$$
0.114708 + 0.993399i $$0.463407\pi$$
$$20$$ 1.00000 0.223607
$$21$$ 0 0
$$22$$ −3.00000 −0.639602
$$23$$ −9.00000 −1.87663 −0.938315 0.345782i $$-0.887614\pi$$
−0.938315 + 0.345782i $$0.887614\pi$$
$$24$$ 0 0
$$25$$ 1.00000 0.200000
$$26$$ 1.00000 0.196116
$$27$$ 0 0
$$28$$ 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
$$33$$ 0 0
$$34$$ −6.00000 −1.02899
$$35$$ 0 0
$$36$$ 0 0
$$37$$ −7.00000 −1.15079 −0.575396 0.817875i $$-0.695152\pi$$
−0.575396 + 0.817875i $$0.695152\pi$$
$$38$$ 1.00000 0.162221
$$39$$ 0 0
$$40$$ 1.00000 0.158114
$$41$$ 3.00000 0.468521 0.234261 0.972174i $$-0.424733\pi$$
0.234261 + 0.972174i $$0.424733\pi$$
$$42$$ 0 0
$$43$$ 2.00000 0.304997 0.152499 0.988304i $$-0.451268\pi$$
0.152499 + 0.988304i $$0.451268\pi$$
$$44$$ −3.00000 −0.452267
$$45$$ 0 0
$$46$$ −9.00000 −1.32698
$$47$$ 9.00000 1.31278 0.656392 0.754420i $$-0.272082\pi$$
0.656392 + 0.754420i $$0.272082\pi$$
$$48$$ 0 0
$$49$$ 0 0
$$50$$ 1.00000 0.141421
$$51$$ 0 0
$$52$$ 1.00000 0.138675
$$53$$ −9.00000 −1.23625 −0.618123 0.786082i $$-0.712106\pi$$
−0.618123 + 0.786082i $$0.712106\pi$$
$$54$$ 0 0
$$55$$ −3.00000 −0.404520
$$56$$ 0 0
$$57$$ 0 0
$$58$$ −6.00000 −0.787839
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ 0 0
$$61$$ −8.00000 −1.02430 −0.512148 0.858898i $$-0.671150\pi$$
−0.512148 + 0.858898i $$0.671150\pi$$
$$62$$ −8.00000 −1.01600
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 1.00000 0.124035
$$66$$ 0 0
$$67$$ 8.00000 0.977356 0.488678 0.872464i $$-0.337479\pi$$
0.488678 + 0.872464i $$0.337479\pi$$
$$68$$ −6.00000 −0.727607
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$72$$ 0 0
$$73$$ 4.00000 0.468165 0.234082 0.972217i $$-0.424791\pi$$
0.234082 + 0.972217i $$0.424791\pi$$
$$74$$ −7.00000 −0.813733
$$75$$ 0 0
$$76$$ 1.00000 0.114708
$$77$$ 0 0
$$78$$ 0 0
$$79$$ −10.0000 −1.12509 −0.562544 0.826767i $$-0.690177\pi$$
−0.562544 + 0.826767i $$0.690177\pi$$
$$80$$ 1.00000 0.111803
$$81$$ 0 0
$$82$$ 3.00000 0.331295
$$83$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$84$$ 0 0
$$85$$ −6.00000 −0.650791
$$86$$ 2.00000 0.215666
$$87$$ 0 0
$$88$$ −3.00000 −0.319801
$$89$$ 6.00000 0.635999 0.317999 0.948091i $$-0.396989\pi$$
0.317999 + 0.948091i $$0.396989\pi$$
$$90$$ 0 0
$$91$$ 0 0
$$92$$ −9.00000 −0.938315
$$93$$ 0 0
$$94$$ 9.00000 0.928279
$$95$$ 1.00000 0.102598
$$96$$ 0 0
$$97$$ 10.0000 1.01535 0.507673 0.861550i $$-0.330506\pi$$
0.507673 + 0.861550i $$0.330506\pi$$
$$98$$ 0 0
$$99$$ 0 0
$$100$$ 1.00000 0.100000
$$101$$ 12.0000 1.19404 0.597022 0.802225i $$-0.296350\pi$$
0.597022 + 0.802225i $$0.296350\pi$$
$$102$$ 0 0
$$103$$ 4.00000 0.394132 0.197066 0.980390i $$-0.436859\pi$$
0.197066 + 0.980390i $$0.436859\pi$$
$$104$$ 1.00000 0.0980581
$$105$$ 0 0
$$106$$ −9.00000 −0.874157
$$107$$ 12.0000 1.16008 0.580042 0.814587i $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ 0 0
$$109$$ −16.0000 −1.53252 −0.766261 0.642529i $$-0.777885\pi$$
−0.766261 + 0.642529i $$0.777885\pi$$
$$110$$ −3.00000 −0.286039
$$111$$ 0 0
$$112$$ 0 0
$$113$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$114$$ 0 0
$$115$$ −9.00000 −0.839254
$$116$$ −6.00000 −0.557086
$$117$$ 0 0
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 0 0
$$121$$ −2.00000 −0.181818
$$122$$ −8.00000 −0.724286
$$123$$ 0 0
$$124$$ −8.00000 −0.718421
$$125$$ 1.00000 0.0894427
$$126$$ 0 0
$$127$$ −1.00000 −0.0887357 −0.0443678 0.999015i $$-0.514127\pi$$
−0.0443678 + 0.999015i $$0.514127\pi$$
$$128$$ 1.00000 0.0883883
$$129$$ 0 0
$$130$$ 1.00000 0.0877058
$$131$$ 3.00000 0.262111 0.131056 0.991375i $$-0.458163\pi$$
0.131056 + 0.991375i $$0.458163\pi$$
$$132$$ 0 0
$$133$$ 0 0
$$134$$ 8.00000 0.691095
$$135$$ 0 0
$$136$$ −6.00000 −0.514496
$$137$$ −12.0000 −1.02523 −0.512615 0.858619i $$-0.671323\pi$$
−0.512615 + 0.858619i $$0.671323\pi$$
$$138$$ 0 0
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 0 0
$$143$$ −3.00000 −0.250873
$$144$$ 0 0
$$145$$ −6.00000 −0.498273
$$146$$ 4.00000 0.331042
$$147$$ 0 0
$$148$$ −7.00000 −0.575396
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ −10.0000 −0.813788 −0.406894 0.913475i $$-0.633388\pi$$
−0.406894 + 0.913475i $$0.633388\pi$$
$$152$$ 1.00000 0.0811107
$$153$$ 0 0
$$154$$ 0 0
$$155$$ −8.00000 −0.642575
$$156$$ 0 0
$$157$$ −23.0000 −1.83560 −0.917800 0.397043i $$-0.870036\pi$$
−0.917800 + 0.397043i $$0.870036\pi$$
$$158$$ −10.0000 −0.795557
$$159$$ 0 0
$$160$$ 1.00000 0.0790569
$$161$$ 0 0
$$162$$ 0 0
$$163$$ 20.0000 1.56652 0.783260 0.621694i $$-0.213555\pi$$
0.783260 + 0.621694i $$0.213555\pi$$
$$164$$ 3.00000 0.234261
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 3.00000 0.232147 0.116073 0.993241i $$-0.462969\pi$$
0.116073 + 0.993241i $$0.462969\pi$$
$$168$$ 0 0
$$169$$ −12.0000 −0.923077
$$170$$ −6.00000 −0.460179
$$171$$ 0 0
$$172$$ 2.00000 0.152499
$$173$$ 9.00000 0.684257 0.342129 0.939653i $$-0.388852\pi$$
0.342129 + 0.939653i $$0.388852\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ −3.00000 −0.226134
$$177$$ 0 0
$$178$$ 6.00000 0.449719
$$179$$ 3.00000 0.224231 0.112115 0.993695i $$-0.464237\pi$$
0.112115 + 0.993695i $$0.464237\pi$$
$$180$$ 0 0
$$181$$ −2.00000 −0.148659 −0.0743294 0.997234i $$-0.523682\pi$$
−0.0743294 + 0.997234i $$0.523682\pi$$
$$182$$ 0 0
$$183$$ 0 0
$$184$$ −9.00000 −0.663489
$$185$$ −7.00000 −0.514650
$$186$$ 0 0
$$187$$ 18.0000 1.31629
$$188$$ 9.00000 0.656392
$$189$$ 0 0
$$190$$ 1.00000 0.0725476
$$191$$ −12.0000 −0.868290 −0.434145 0.900843i $$-0.642949\pi$$
−0.434145 + 0.900843i $$0.642949\pi$$
$$192$$ 0 0
$$193$$ −16.0000 −1.15171 −0.575853 0.817554i $$-0.695330\pi$$
−0.575853 + 0.817554i $$0.695330\pi$$
$$194$$ 10.0000 0.717958
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −15.0000 −1.06871 −0.534353 0.845262i $$-0.679445\pi$$
−0.534353 + 0.845262i $$0.679445\pi$$
$$198$$ 0 0
$$199$$ 16.0000 1.13421 0.567105 0.823646i $$-0.308063\pi$$
0.567105 + 0.823646i $$0.308063\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ 0 0
$$202$$ 12.0000 0.844317
$$203$$ 0 0
$$204$$ 0 0
$$205$$ 3.00000 0.209529
$$206$$ 4.00000 0.278693
$$207$$ 0 0
$$208$$ 1.00000 0.0693375
$$209$$ −3.00000 −0.207514
$$210$$ 0 0
$$211$$ 23.0000 1.58339 0.791693 0.610920i $$-0.209200\pi$$
0.791693 + 0.610920i $$0.209200\pi$$
$$212$$ −9.00000 −0.618123
$$213$$ 0 0
$$214$$ 12.0000 0.820303
$$215$$ 2.00000 0.136399
$$216$$ 0 0
$$217$$ 0 0
$$218$$ −16.0000 −1.08366
$$219$$ 0 0
$$220$$ −3.00000 −0.202260
$$221$$ −6.00000 −0.403604
$$222$$ 0 0
$$223$$ −8.00000 −0.535720 −0.267860 0.963458i $$-0.586316\pi$$
−0.267860 + 0.963458i $$0.586316\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ 0 0
$$229$$ 4.00000 0.264327 0.132164 0.991228i $$-0.457808\pi$$
0.132164 + 0.991228i $$0.457808\pi$$
$$230$$ −9.00000 −0.593442
$$231$$ 0 0
$$232$$ −6.00000 −0.393919
$$233$$ 6.00000 0.393073 0.196537 0.980497i $$-0.437031\pi$$
0.196537 + 0.980497i $$0.437031\pi$$
$$234$$ 0 0
$$235$$ 9.00000 0.587095
$$236$$ 0 0
$$237$$ 0 0
$$238$$ 0 0
$$239$$ 6.00000 0.388108 0.194054 0.980991i $$-0.437836\pi$$
0.194054 + 0.980991i $$0.437836\pi$$
$$240$$ 0 0
$$241$$ 1.00000 0.0644157 0.0322078 0.999481i $$-0.489746\pi$$
0.0322078 + 0.999481i $$0.489746\pi$$
$$242$$ −2.00000 −0.128565
$$243$$ 0 0
$$244$$ −8.00000 −0.512148
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 1.00000 0.0636285
$$248$$ −8.00000 −0.508001
$$249$$ 0 0
$$250$$ 1.00000 0.0632456
$$251$$ −15.0000 −0.946792 −0.473396 0.880850i $$-0.656972\pi$$
−0.473396 + 0.880850i $$0.656972\pi$$
$$252$$ 0 0
$$253$$ 27.0000 1.69748
$$254$$ −1.00000 −0.0627456
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$258$$ 0 0
$$259$$ 0 0
$$260$$ 1.00000 0.0620174
$$261$$ 0 0
$$262$$ 3.00000 0.185341
$$263$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$264$$ 0 0
$$265$$ −9.00000 −0.552866
$$266$$ 0 0
$$267$$ 0 0
$$268$$ 8.00000 0.488678
$$269$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$270$$ 0 0
$$271$$ 16.0000 0.971931 0.485965 0.873978i $$-0.338468\pi$$
0.485965 + 0.873978i $$0.338468\pi$$
$$272$$ −6.00000 −0.363803
$$273$$ 0 0
$$274$$ −12.0000 −0.724947
$$275$$ −3.00000 −0.180907
$$276$$ 0 0
$$277$$ −10.0000 −0.600842 −0.300421 0.953807i $$-0.597127\pi$$
−0.300421 + 0.953807i $$0.597127\pi$$
$$278$$ 4.00000 0.239904
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 27.0000 1.61068 0.805342 0.592810i $$-0.201981\pi$$
0.805342 + 0.592810i $$0.201981\pi$$
$$282$$ 0 0
$$283$$ −14.0000 −0.832214 −0.416107 0.909316i $$-0.636606\pi$$
−0.416107 + 0.909316i $$0.636606\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ −3.00000 −0.177394
$$287$$ 0 0
$$288$$ 0 0
$$289$$ 19.0000 1.11765
$$290$$ −6.00000 −0.352332
$$291$$ 0 0
$$292$$ 4.00000 0.234082
$$293$$ −9.00000 −0.525786 −0.262893 0.964825i $$-0.584677\pi$$
−0.262893 + 0.964825i $$0.584677\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −7.00000 −0.406867
$$297$$ 0 0
$$298$$ 6.00000 0.347571
$$299$$ −9.00000 −0.520483
$$300$$ 0 0
$$301$$ 0 0
$$302$$ −10.0000 −0.575435
$$303$$ 0 0
$$304$$ 1.00000 0.0573539
$$305$$ −8.00000 −0.458079
$$306$$ 0 0
$$307$$ −14.0000 −0.799022 −0.399511 0.916728i $$-0.630820\pi$$
−0.399511 + 0.916728i $$0.630820\pi$$
$$308$$ 0 0
$$309$$ 0 0
$$310$$ −8.00000 −0.454369
$$311$$ −24.0000 −1.36092 −0.680458 0.732787i $$-0.738219\pi$$
−0.680458 + 0.732787i $$0.738219\pi$$
$$312$$ 0 0
$$313$$ 28.0000 1.58265 0.791327 0.611393i $$-0.209391\pi$$
0.791327 + 0.611393i $$0.209391\pi$$
$$314$$ −23.0000 −1.29797
$$315$$ 0 0
$$316$$ −10.0000 −0.562544
$$317$$ −6.00000 −0.336994 −0.168497 0.985702i $$-0.553891\pi$$
−0.168497 + 0.985702i $$0.553891\pi$$
$$318$$ 0 0
$$319$$ 18.0000 1.00781
$$320$$ 1.00000 0.0559017
$$321$$ 0 0
$$322$$ 0 0
$$323$$ −6.00000 −0.333849
$$324$$ 0 0
$$325$$ 1.00000 0.0554700
$$326$$ 20.0000 1.10770
$$327$$ 0 0
$$328$$ 3.00000 0.165647
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −7.00000 −0.384755 −0.192377 0.981321i $$-0.561620\pi$$
−0.192377 + 0.981321i $$0.561620\pi$$
$$332$$ 0 0
$$333$$ 0 0
$$334$$ 3.00000 0.164153
$$335$$ 8.00000 0.437087
$$336$$ 0 0
$$337$$ −22.0000 −1.19842 −0.599208 0.800593i $$-0.704518\pi$$
−0.599208 + 0.800593i $$0.704518\pi$$
$$338$$ −12.0000 −0.652714
$$339$$ 0 0
$$340$$ −6.00000 −0.325396
$$341$$ 24.0000 1.29967
$$342$$ 0 0
$$343$$ 0 0
$$344$$ 2.00000 0.107833
$$345$$ 0 0
$$346$$ 9.00000 0.483843
$$347$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$348$$ 0 0
$$349$$ −26.0000 −1.39175 −0.695874 0.718164i $$-0.744983\pi$$
−0.695874 + 0.718164i $$0.744983\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −3.00000 −0.159901
$$353$$ 12.0000 0.638696 0.319348 0.947638i $$-0.396536\pi$$
0.319348 + 0.947638i $$0.396536\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 6.00000 0.317999
$$357$$ 0 0
$$358$$ 3.00000 0.158555
$$359$$ −18.0000 −0.950004 −0.475002 0.879985i $$-0.657553\pi$$
−0.475002 + 0.879985i $$0.657553\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ −2.00000 −0.105118
$$363$$ 0 0
$$364$$ 0 0
$$365$$ 4.00000 0.209370
$$366$$ 0 0
$$367$$ 19.0000 0.991792 0.495896 0.868382i $$-0.334840\pi$$
0.495896 + 0.868382i $$0.334840\pi$$
$$368$$ −9.00000 −0.469157
$$369$$ 0 0
$$370$$ −7.00000 −0.363913
$$371$$ 0 0
$$372$$ 0 0
$$373$$ 2.00000 0.103556 0.0517780 0.998659i $$-0.483511\pi$$
0.0517780 + 0.998659i $$0.483511\pi$$
$$374$$ 18.0000 0.930758
$$375$$ 0 0
$$376$$ 9.00000 0.464140
$$377$$ −6.00000 −0.309016
$$378$$ 0 0
$$379$$ 23.0000 1.18143 0.590715 0.806880i $$-0.298846\pi$$
0.590715 + 0.806880i $$0.298846\pi$$
$$380$$ 1.00000 0.0512989
$$381$$ 0 0
$$382$$ −12.0000 −0.613973
$$383$$ 21.0000 1.07305 0.536525 0.843884i $$-0.319737\pi$$
0.536525 + 0.843884i $$0.319737\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −16.0000 −0.814379
$$387$$ 0 0
$$388$$ 10.0000 0.507673
$$389$$ 12.0000 0.608424 0.304212 0.952604i $$-0.401607\pi$$
0.304212 + 0.952604i $$0.401607\pi$$
$$390$$ 0 0
$$391$$ 54.0000 2.73090
$$392$$ 0 0
$$393$$ 0 0
$$394$$ −15.0000 −0.755689
$$395$$ −10.0000 −0.503155
$$396$$ 0 0
$$397$$ −14.0000 −0.702640 −0.351320 0.936255i $$-0.614267\pi$$
−0.351320 + 0.936255i $$0.614267\pi$$
$$398$$ 16.0000 0.802008
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ 27.0000 1.34832 0.674158 0.738587i $$-0.264507\pi$$
0.674158 + 0.738587i $$0.264507\pi$$
$$402$$ 0 0
$$403$$ −8.00000 −0.398508
$$404$$ 12.0000 0.597022
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 21.0000 1.04093
$$408$$ 0 0
$$409$$ −26.0000 −1.28562 −0.642809 0.766027i $$-0.722231\pi$$
−0.642809 + 0.766027i $$0.722231\pi$$
$$410$$ 3.00000 0.148159
$$411$$ 0 0
$$412$$ 4.00000 0.197066
$$413$$ 0 0
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 1.00000 0.0490290
$$417$$ 0 0
$$418$$ −3.00000 −0.146735
$$419$$ −9.00000 −0.439679 −0.219839 0.975536i $$-0.570553\pi$$
−0.219839 + 0.975536i $$0.570553\pi$$
$$420$$ 0 0
$$421$$ 2.00000 0.0974740 0.0487370 0.998812i $$-0.484480\pi$$
0.0487370 + 0.998812i $$0.484480\pi$$
$$422$$ 23.0000 1.11962
$$423$$ 0 0
$$424$$ −9.00000 −0.437079
$$425$$ −6.00000 −0.291043
$$426$$ 0 0
$$427$$ 0 0
$$428$$ 12.0000 0.580042
$$429$$ 0 0
$$430$$ 2.00000 0.0964486
$$431$$ −12.0000 −0.578020 −0.289010 0.957326i $$-0.593326\pi$$
−0.289010 + 0.957326i $$0.593326\pi$$
$$432$$ 0 0
$$433$$ 40.0000 1.92228 0.961139 0.276066i $$-0.0890309\pi$$
0.961139 + 0.276066i $$0.0890309\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ −16.0000 −0.766261
$$437$$ −9.00000 −0.430528
$$438$$ 0 0
$$439$$ −26.0000 −1.24091 −0.620456 0.784241i $$-0.713053\pi$$
−0.620456 + 0.784241i $$0.713053\pi$$
$$440$$ −3.00000 −0.143019
$$441$$ 0 0
$$442$$ −6.00000 −0.285391
$$443$$ −12.0000 −0.570137 −0.285069 0.958507i $$-0.592016\pi$$
−0.285069 + 0.958507i $$0.592016\pi$$
$$444$$ 0 0
$$445$$ 6.00000 0.284427
$$446$$ −8.00000 −0.378811
$$447$$ 0 0
$$448$$ 0 0
$$449$$ −21.0000 −0.991051 −0.495526 0.868593i $$-0.665025\pi$$
−0.495526 + 0.868593i $$0.665025\pi$$
$$450$$ 0 0
$$451$$ −9.00000 −0.423793
$$452$$ 0 0
$$453$$ 0 0
$$454$$ −12.0000 −0.563188
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 14.0000 0.654892 0.327446 0.944870i $$-0.393812\pi$$
0.327446 + 0.944870i $$0.393812\pi$$
$$458$$ 4.00000 0.186908
$$459$$ 0 0
$$460$$ −9.00000 −0.419627
$$461$$ −30.0000 −1.39724 −0.698620 0.715493i $$-0.746202\pi$$
−0.698620 + 0.715493i $$0.746202\pi$$
$$462$$ 0 0
$$463$$ −1.00000 −0.0464739 −0.0232370 0.999730i $$-0.507397\pi$$
−0.0232370 + 0.999730i $$0.507397\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 0 0
$$466$$ 6.00000 0.277945
$$467$$ −6.00000 −0.277647 −0.138823 0.990317i $$-0.544332\pi$$
−0.138823 + 0.990317i $$0.544332\pi$$
$$468$$ 0 0
$$469$$ 0 0
$$470$$ 9.00000 0.415139
$$471$$ 0 0
$$472$$ 0 0
$$473$$ −6.00000 −0.275880
$$474$$ 0 0
$$475$$ 1.00000 0.0458831
$$476$$ 0 0
$$477$$ 0 0
$$478$$ 6.00000 0.274434
$$479$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$480$$ 0 0
$$481$$ −7.00000 −0.319173
$$482$$ 1.00000 0.0455488
$$483$$ 0 0
$$484$$ −2.00000 −0.0909091
$$485$$ 10.0000 0.454077
$$486$$ 0 0
$$487$$ −16.0000 −0.725029 −0.362515 0.931978i $$-0.618082\pi$$
−0.362515 + 0.931978i $$0.618082\pi$$
$$488$$ −8.00000 −0.362143
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −36.0000 −1.62466 −0.812329 0.583200i $$-0.801800\pi$$
−0.812329 + 0.583200i $$0.801800\pi$$
$$492$$ 0 0
$$493$$ 36.0000 1.62136
$$494$$ 1.00000 0.0449921
$$495$$ 0 0
$$496$$ −8.00000 −0.359211
$$497$$ 0 0
$$498$$ 0 0
$$499$$ −4.00000 −0.179065 −0.0895323 0.995984i $$-0.528537\pi$$
−0.0895323 + 0.995984i $$0.528537\pi$$
$$500$$ 1.00000 0.0447214
$$501$$ 0 0
$$502$$ −15.0000 −0.669483
$$503$$ 24.0000 1.07011 0.535054 0.844818i $$-0.320291\pi$$
0.535054 + 0.844818i $$0.320291\pi$$
$$504$$ 0 0
$$505$$ 12.0000 0.533993
$$506$$ 27.0000 1.20030
$$507$$ 0 0
$$508$$ −1.00000 −0.0443678
$$509$$ −42.0000 −1.86162 −0.930809 0.365507i $$-0.880896\pi$$
−0.930809 + 0.365507i $$0.880896\pi$$
$$510$$ 0 0
$$511$$ 0 0
$$512$$ 1.00000 0.0441942
$$513$$ 0 0
$$514$$ 0 0
$$515$$ 4.00000 0.176261
$$516$$ 0 0
$$517$$ −27.0000 −1.18746
$$518$$ 0 0
$$519$$ 0 0
$$520$$ 1.00000 0.0438529
$$521$$ −15.0000 −0.657162 −0.328581 0.944476i $$-0.606570\pi$$
−0.328581 + 0.944476i $$0.606570\pi$$
$$522$$ 0 0
$$523$$ 28.0000 1.22435 0.612177 0.790721i $$-0.290294\pi$$
0.612177 + 0.790721i $$0.290294\pi$$
$$524$$ 3.00000 0.131056
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 48.0000 2.09091
$$528$$ 0 0
$$529$$ 58.0000 2.52174
$$530$$ −9.00000 −0.390935
$$531$$ 0 0
$$532$$ 0 0
$$533$$ 3.00000 0.129944
$$534$$ 0 0
$$535$$ 12.0000 0.518805
$$536$$ 8.00000 0.345547
$$537$$ 0 0
$$538$$ 0 0
$$539$$ 0 0
$$540$$ 0 0
$$541$$ 8.00000 0.343947 0.171973 0.985102i $$-0.444986\pi$$
0.171973 + 0.985102i $$0.444986\pi$$
$$542$$ 16.0000 0.687259
$$543$$ 0 0
$$544$$ −6.00000 −0.257248
$$545$$ −16.0000 −0.685365
$$546$$ 0 0
$$547$$ 8.00000 0.342055 0.171028 0.985266i $$-0.445291\pi$$
0.171028 + 0.985266i $$0.445291\pi$$
$$548$$ −12.0000 −0.512615
$$549$$ 0 0
$$550$$ −3.00000 −0.127920
$$551$$ −6.00000 −0.255609
$$552$$ 0 0
$$553$$ 0 0
$$554$$ −10.0000 −0.424859
$$555$$ 0 0
$$556$$ 4.00000 0.169638
$$557$$ 9.00000 0.381342 0.190671 0.981654i $$-0.438934\pi$$
0.190671 + 0.981654i $$0.438934\pi$$
$$558$$ 0 0
$$559$$ 2.00000 0.0845910
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 27.0000 1.13893
$$563$$ 42.0000 1.77009 0.885044 0.465506i $$-0.154128\pi$$
0.885044 + 0.465506i $$0.154128\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −14.0000 −0.588464
$$567$$ 0 0
$$568$$ 0 0
$$569$$ −21.0000 −0.880366 −0.440183 0.897908i $$-0.645086\pi$$
−0.440183 + 0.897908i $$0.645086\pi$$
$$570$$ 0 0
$$571$$ 20.0000 0.836974 0.418487 0.908223i $$-0.362561\pi$$
0.418487 + 0.908223i $$0.362561\pi$$
$$572$$ −3.00000 −0.125436
$$573$$ 0 0
$$574$$ 0 0
$$575$$ −9.00000 −0.375326
$$576$$ 0 0
$$577$$ −44.0000 −1.83174 −0.915872 0.401470i $$-0.868499\pi$$
−0.915872 + 0.401470i $$0.868499\pi$$
$$578$$ 19.0000 0.790296
$$579$$ 0 0
$$580$$ −6.00000 −0.249136
$$581$$ 0 0
$$582$$ 0 0
$$583$$ 27.0000 1.11823
$$584$$ 4.00000 0.165521
$$585$$ 0 0
$$586$$ −9.00000 −0.371787
$$587$$ −24.0000 −0.990586 −0.495293 0.868726i $$-0.664939\pi$$
−0.495293 + 0.868726i $$0.664939\pi$$
$$588$$ 0 0
$$589$$ −8.00000 −0.329634
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −7.00000 −0.287698
$$593$$ 24.0000 0.985562 0.492781 0.870153i $$-0.335980\pi$$
0.492781 + 0.870153i $$0.335980\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 6.00000 0.245770
$$597$$ 0 0
$$598$$ −9.00000 −0.368037
$$599$$ 42.0000 1.71607 0.858037 0.513588i $$-0.171684\pi$$
0.858037 + 0.513588i $$0.171684\pi$$
$$600$$ 0 0
$$601$$ −26.0000 −1.06056 −0.530281 0.847822i $$-0.677914\pi$$
−0.530281 + 0.847822i $$0.677914\pi$$
$$602$$ 0 0
$$603$$ 0 0
$$604$$ −10.0000 −0.406894
$$605$$ −2.00000 −0.0813116
$$606$$ 0 0
$$607$$ 1.00000 0.0405887 0.0202944 0.999794i $$-0.493540\pi$$
0.0202944 + 0.999794i $$0.493540\pi$$
$$608$$ 1.00000 0.0405554
$$609$$ 0 0
$$610$$ −8.00000 −0.323911
$$611$$ 9.00000 0.364101
$$612$$ 0 0
$$613$$ 29.0000 1.17130 0.585649 0.810564i $$-0.300840\pi$$
0.585649 + 0.810564i $$0.300840\pi$$
$$614$$ −14.0000 −0.564994
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 18.0000 0.724653 0.362326 0.932051i $$-0.381983\pi$$
0.362326 + 0.932051i $$0.381983\pi$$
$$618$$ 0 0
$$619$$ −23.0000 −0.924448 −0.462224 0.886763i $$-0.652948\pi$$
−0.462224 + 0.886763i $$0.652948\pi$$
$$620$$ −8.00000 −0.321288
$$621$$ 0 0
$$622$$ −24.0000 −0.962312
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 1.00000 0.0400000
$$626$$ 28.0000 1.11911
$$627$$ 0 0
$$628$$ −23.0000 −0.917800
$$629$$ 42.0000 1.67465
$$630$$ 0 0
$$631$$ 20.0000 0.796187 0.398094 0.917345i $$-0.369672\pi$$
0.398094 + 0.917345i $$0.369672\pi$$
$$632$$ −10.0000 −0.397779
$$633$$ 0 0
$$634$$ −6.00000 −0.238290
$$635$$ −1.00000 −0.0396838
$$636$$ 0 0
$$637$$ 0 0
$$638$$ 18.0000 0.712627
$$639$$ 0 0
$$640$$ 1.00000 0.0395285
$$641$$ −27.0000 −1.06644 −0.533218 0.845978i $$-0.679017\pi$$
−0.533218 + 0.845978i $$0.679017\pi$$
$$642$$ 0 0
$$643$$ −2.00000 −0.0788723 −0.0394362 0.999222i $$-0.512556\pi$$
−0.0394362 + 0.999222i $$0.512556\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ −6.00000 −0.236067
$$647$$ 33.0000 1.29736 0.648682 0.761060i $$-0.275321\pi$$
0.648682 + 0.761060i $$0.275321\pi$$
$$648$$ 0 0
$$649$$ 0 0
$$650$$ 1.00000 0.0392232
$$651$$ 0 0
$$652$$ 20.0000 0.783260
$$653$$ 9.00000 0.352197 0.176099 0.984373i $$-0.443652\pi$$
0.176099 + 0.984373i $$0.443652\pi$$
$$654$$ 0 0
$$655$$ 3.00000 0.117220
$$656$$ 3.00000 0.117130
$$657$$ 0 0
$$658$$ 0 0
$$659$$ 24.0000 0.934907 0.467454 0.884018i $$-0.345171\pi$$
0.467454 + 0.884018i $$0.345171\pi$$
$$660$$ 0 0
$$661$$ 28.0000 1.08907 0.544537 0.838737i $$-0.316705\pi$$
0.544537 + 0.838737i $$0.316705\pi$$
$$662$$ −7.00000 −0.272063
$$663$$ 0 0
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 54.0000 2.09089
$$668$$ 3.00000 0.116073
$$669$$ 0 0
$$670$$ 8.00000 0.309067
$$671$$ 24.0000 0.926510
$$672$$ 0 0
$$673$$ −34.0000 −1.31060 −0.655302 0.755367i $$-0.727459\pi$$
−0.655302 + 0.755367i $$0.727459\pi$$
$$674$$ −22.0000 −0.847408
$$675$$ 0 0
$$676$$ −12.0000 −0.461538
$$677$$ −9.00000 −0.345898 −0.172949 0.984931i $$-0.555330\pi$$
−0.172949 + 0.984931i $$0.555330\pi$$
$$678$$ 0 0
$$679$$ 0 0
$$680$$ −6.00000 −0.230089
$$681$$ 0 0
$$682$$ 24.0000 0.919007
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ 0 0
$$685$$ −12.0000 −0.458496
$$686$$ 0 0
$$687$$ 0 0
$$688$$ 2.00000 0.0762493
$$689$$ −9.00000 −0.342873
$$690$$ 0 0
$$691$$ −32.0000 −1.21734 −0.608669 0.793424i $$-0.708296\pi$$
−0.608669 + 0.793424i $$0.708296\pi$$
$$692$$ 9.00000 0.342129
$$693$$ 0 0
$$694$$ 0 0
$$695$$ 4.00000 0.151729
$$696$$ 0 0
$$697$$ −18.0000 −0.681799
$$698$$ −26.0000 −0.984115
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 30.0000 1.13308 0.566542 0.824033i $$-0.308281\pi$$
0.566542 + 0.824033i $$0.308281\pi$$
$$702$$ 0 0
$$703$$ −7.00000 −0.264010
$$704$$ −3.00000 −0.113067
$$705$$ 0 0
$$706$$ 12.0000 0.451626
$$707$$ 0 0
$$708$$ 0 0
$$709$$ −46.0000 −1.72757 −0.863783 0.503864i $$-0.831911\pi$$
−0.863783 + 0.503864i $$0.831911\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 6.00000 0.224860
$$713$$ 72.0000 2.69642
$$714$$ 0 0
$$715$$ −3.00000 −0.112194
$$716$$ 3.00000 0.112115
$$717$$ 0 0
$$718$$ −18.0000 −0.671754
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ 0 0
$$722$$ −18.0000 −0.669891
$$723$$ 0 0
$$724$$ −2.00000 −0.0743294
$$725$$ −6.00000 −0.222834
$$726$$ 0 0
$$727$$ 1.00000 0.0370879 0.0185440 0.999828i $$-0.494097\pi$$
0.0185440 + 0.999828i $$0.494097\pi$$
$$728$$ 0 0
$$729$$ 0 0
$$730$$ 4.00000 0.148047
$$731$$ −12.0000 −0.443836
$$732$$ 0 0
$$733$$ 43.0000 1.58824 0.794121 0.607760i $$-0.207932\pi$$
0.794121 + 0.607760i $$0.207932\pi$$
$$734$$ 19.0000 0.701303
$$735$$ 0 0
$$736$$ −9.00000 −0.331744
$$737$$ −24.0000 −0.884051
$$738$$ 0 0
$$739$$ 35.0000 1.28750 0.643748 0.765238i $$-0.277379\pi$$
0.643748 + 0.765238i $$0.277379\pi$$
$$740$$ −7.00000 −0.257325
$$741$$ 0 0
$$742$$ 0 0
$$743$$ 45.0000 1.65089 0.825445 0.564483i $$-0.190924\pi$$
0.825445 + 0.564483i $$0.190924\pi$$
$$744$$ 0 0
$$745$$ 6.00000 0.219823
$$746$$ 2.00000 0.0732252
$$747$$ 0 0
$$748$$ 18.0000 0.658145
$$749$$ 0 0
$$750$$ 0 0
$$751$$ −10.0000 −0.364905 −0.182453 0.983215i $$-0.558404\pi$$
−0.182453 + 0.983215i $$0.558404\pi$$
$$752$$ 9.00000 0.328196
$$753$$ 0 0
$$754$$ −6.00000 −0.218507
$$755$$ −10.0000 −0.363937
$$756$$ 0 0
$$757$$ 38.0000 1.38113 0.690567 0.723269i $$-0.257361\pi$$
0.690567 + 0.723269i $$0.257361\pi$$
$$758$$ 23.0000 0.835398
$$759$$ 0 0
$$760$$ 1.00000 0.0362738
$$761$$ −27.0000 −0.978749 −0.489375 0.872074i $$-0.662775\pi$$
−0.489375 + 0.872074i $$0.662775\pi$$
$$762$$ 0 0
$$763$$ 0 0
$$764$$ −12.0000 −0.434145
$$765$$ 0 0
$$766$$ 21.0000 0.758761
$$767$$ 0 0
$$768$$ 0 0
$$769$$ −23.0000 −0.829401 −0.414701 0.909958i $$-0.636114\pi$$
−0.414701 + 0.909958i $$0.636114\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −16.0000 −0.575853
$$773$$ −51.0000 −1.83434 −0.917171 0.398493i $$-0.869533\pi$$
−0.917171 + 0.398493i $$0.869533\pi$$
$$774$$ 0 0
$$775$$ −8.00000 −0.287368
$$776$$ 10.0000 0.358979
$$777$$ 0 0
$$778$$ 12.0000 0.430221
$$779$$ 3.00000 0.107486
$$780$$ 0 0
$$781$$ 0 0
$$782$$ 54.0000 1.93104
$$783$$ 0 0
$$784$$ 0 0
$$785$$ −23.0000 −0.820905
$$786$$ 0 0
$$787$$ 22.0000 0.784215 0.392108 0.919919i $$-0.371746\pi$$
0.392108 + 0.919919i $$0.371746\pi$$
$$788$$ −15.0000 −0.534353
$$789$$ 0 0
$$790$$ −10.0000 −0.355784
$$791$$ 0 0
$$792$$ 0 0
$$793$$ −8.00000 −0.284088
$$794$$ −14.0000 −0.496841
$$795$$ 0 0
$$796$$ 16.0000 0.567105
$$797$$ 6.00000 0.212531 0.106265 0.994338i $$-0.466111\pi$$
0.106265 + 0.994338i $$0.466111\pi$$
$$798$$ 0 0
$$799$$ −54.0000 −1.91038
$$800$$ 1.00000 0.0353553
$$801$$ 0 0
$$802$$ 27.0000 0.953403
$$803$$ −12.0000 −0.423471
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −8.00000 −0.281788
$$807$$ 0 0
$$808$$ 12.0000 0.422159
$$809$$ 9.00000 0.316423 0.158212 0.987405i $$-0.449427\pi$$
0.158212 + 0.987405i $$0.449427\pi$$
$$810$$ 0 0
$$811$$ 25.0000 0.877869 0.438934 0.898519i $$-0.355356\pi$$
0.438934 + 0.898519i $$0.355356\pi$$
$$812$$ 0 0
$$813$$ 0 0
$$814$$ 21.0000 0.736050
$$815$$ 20.0000 0.700569
$$816$$ 0 0
$$817$$ 2.00000 0.0699711
$$818$$ −26.0000 −0.909069
$$819$$ 0 0
$$820$$ 3.00000 0.104765
$$821$$ −30.0000 −1.04701 −0.523504 0.852023i $$-0.675375\pi$$
−0.523504 + 0.852023i $$0.675375\pi$$
$$822$$ 0 0
$$823$$ −4.00000 −0.139431 −0.0697156 0.997567i $$-0.522209\pi$$
−0.0697156 + 0.997567i $$0.522209\pi$$
$$824$$ 4.00000 0.139347
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −6.00000 −0.208640 −0.104320 0.994544i $$-0.533267\pi$$
−0.104320 + 0.994544i $$0.533267\pi$$
$$828$$ 0 0
$$829$$ −14.0000 −0.486240 −0.243120 0.969996i $$-0.578171\pi$$
−0.243120 + 0.969996i $$0.578171\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 1.00000 0.0346688
$$833$$ 0 0
$$834$$ 0 0
$$835$$ 3.00000 0.103819
$$836$$ −3.00000 −0.103757
$$837$$ 0 0
$$838$$ −9.00000 −0.310900
$$839$$ −30.0000 −1.03572 −0.517858 0.855467i $$-0.673270\pi$$
−0.517858 + 0.855467i $$0.673270\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 2.00000 0.0689246
$$843$$ 0 0
$$844$$ 23.0000 0.791693
$$845$$ −12.0000 −0.412813
$$846$$ 0 0
$$847$$ 0 0
$$848$$ −9.00000 −0.309061
$$849$$ 0 0
$$850$$ −6.00000 −0.205798
$$851$$ 63.0000 2.15961
$$852$$ 0 0
$$853$$ 19.0000 0.650548 0.325274 0.945620i $$-0.394544\pi$$
0.325274 + 0.945620i $$0.394544\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ −18.0000 −0.614868 −0.307434 0.951569i $$-0.599470\pi$$
−0.307434 + 0.951569i $$0.599470\pi$$
$$858$$ 0 0
$$859$$ −32.0000 −1.09183 −0.545913 0.837842i $$-0.683817\pi$$
−0.545913 + 0.837842i $$0.683817\pi$$
$$860$$ 2.00000 0.0681994
$$861$$ 0 0
$$862$$ −12.0000 −0.408722
$$863$$ 3.00000 0.102121 0.0510606 0.998696i $$-0.483740\pi$$
0.0510606 + 0.998696i $$0.483740\pi$$
$$864$$ 0 0
$$865$$ 9.00000 0.306009
$$866$$ 40.0000 1.35926
$$867$$ 0 0
$$868$$ 0 0
$$869$$ 30.0000 1.01768
$$870$$ 0 0
$$871$$ 8.00000 0.271070
$$872$$ −16.0000 −0.541828
$$873$$ 0 0
$$874$$ −9.00000 −0.304430
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −13.0000 −0.438979 −0.219489 0.975615i $$-0.570439\pi$$
−0.219489 + 0.975615i $$0.570439\pi$$
$$878$$ −26.0000 −0.877457
$$879$$ 0 0
$$880$$ −3.00000 −0.101130
$$881$$ 33.0000 1.11180 0.555899 0.831250i $$-0.312374\pi$$
0.555899 + 0.831250i $$0.312374\pi$$
$$882$$ 0 0
$$883$$ 8.00000 0.269221 0.134611 0.990899i $$-0.457022\pi$$
0.134611 + 0.990899i $$0.457022\pi$$
$$884$$ −6.00000 −0.201802
$$885$$ 0 0
$$886$$ −12.0000 −0.403148
$$887$$ −48.0000 −1.61168 −0.805841 0.592132i $$-0.798286\pi$$
−0.805841 + 0.592132i $$0.798286\pi$$
$$888$$ 0 0
$$889$$ 0 0
$$890$$ 6.00000 0.201120
$$891$$ 0 0
$$892$$ −8.00000 −0.267860
$$893$$ 9.00000 0.301174
$$894$$ 0 0
$$895$$ 3.00000 0.100279
$$896$$ 0 0
$$897$$ 0 0
$$898$$ −21.0000 −0.700779
$$899$$ 48.0000 1.60089
$$900$$ 0 0
$$901$$ 54.0000 1.79900
$$902$$ −9.00000 −0.299667
$$903$$ 0 0
$$904$$ 0 0
$$905$$ −2.00000 −0.0664822
$$906$$ 0 0
$$907$$ −10.0000 −0.332045 −0.166022 0.986122i $$-0.553092\pi$$
−0.166022 + 0.986122i $$0.553092\pi$$
$$908$$ −12.0000 −0.398234
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 30.0000 0.993944 0.496972 0.867766i $$-0.334445\pi$$
0.496972 + 0.867766i $$0.334445\pi$$
$$912$$ 0 0
$$913$$ 0 0
$$914$$ 14.0000 0.463079
$$915$$ 0 0
$$916$$ 4.00000 0.132164
$$917$$ 0 0
$$918$$ 0 0
$$919$$ −22.0000 −0.725713 −0.362857 0.931845i $$-0.618198\pi$$
−0.362857 + 0.931845i $$0.618198\pi$$
$$920$$ −9.00000 −0.296721
$$921$$ 0 0
$$922$$ −30.0000 −0.987997
$$923$$ 0 0
$$924$$ 0 0
$$925$$ −7.00000 −0.230159
$$926$$ −1.00000 −0.0328620
$$927$$ 0 0
$$928$$ −6.00000 −0.196960
$$929$$ 57.0000 1.87011 0.935055 0.354504i $$-0.115350\pi$$
0.935055 + 0.354504i $$0.115350\pi$$
$$930$$ 0 0
$$931$$ 0 0
$$932$$ 6.00000 0.196537
$$933$$ 0 0
$$934$$ −6.00000 −0.196326
$$935$$ 18.0000 0.588663
$$936$$ 0 0
$$937$$ 10.0000 0.326686 0.163343 0.986569i $$-0.447772\pi$$
0.163343 + 0.986569i $$0.447772\pi$$
$$938$$ 0 0
$$939$$ 0 0
$$940$$ 9.00000 0.293548
$$941$$ 48.0000 1.56476 0.782378 0.622804i $$-0.214007\pi$$
0.782378 + 0.622804i $$0.214007\pi$$
$$942$$ 0 0
$$943$$ −27.0000 −0.879241
$$944$$ 0 0
$$945$$ 0 0
$$946$$ −6.00000 −0.195077
$$947$$ −6.00000 −0.194974 −0.0974869 0.995237i $$-0.531080\pi$$
−0.0974869 + 0.995237i $$0.531080\pi$$
$$948$$ 0 0
$$949$$ 4.00000 0.129845
$$950$$ 1.00000 0.0324443
$$951$$ 0 0
$$952$$ 0 0
$$953$$ −36.0000 −1.16615 −0.583077 0.812417i $$-0.698151\pi$$
−0.583077 + 0.812417i $$0.698151\pi$$
$$954$$ 0 0
$$955$$ −12.0000 −0.388311
$$956$$ 6.00000 0.194054
$$957$$ 0 0
$$958$$ 0 0
$$959$$ 0 0
$$960$$ 0 0
$$961$$ 33.0000 1.06452
$$962$$ −7.00000 −0.225689
$$963$$ 0 0
$$964$$ 1.00000 0.0322078
$$965$$ −16.0000 −0.515058
$$966$$ 0 0
$$967$$ 32.0000 1.02905 0.514525 0.857475i $$-0.327968\pi$$
0.514525 + 0.857475i $$0.327968\pi$$
$$968$$ −2.00000 −0.0642824
$$969$$ 0 0
$$970$$ 10.0000 0.321081
$$971$$ −45.0000 −1.44412 −0.722059 0.691831i $$-0.756804\pi$$
−0.722059 + 0.691831i $$0.756804\pi$$
$$972$$ 0 0
$$973$$ 0 0
$$974$$ −16.0000 −0.512673
$$975$$ 0 0
$$976$$ −8.00000 −0.256074
$$977$$ 42.0000 1.34370 0.671850 0.740688i $$-0.265500\pi$$
0.671850 + 0.740688i $$0.265500\pi$$
$$978$$ 0 0
$$979$$ −18.0000 −0.575282
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −36.0000 −1.14881
$$983$$ 3.00000 0.0956851 0.0478426 0.998855i $$-0.484765\pi$$
0.0478426 + 0.998855i $$0.484765\pi$$
$$984$$ 0 0
$$985$$ −15.0000 −0.477940
$$986$$ 36.0000 1.14647
$$987$$ 0 0
$$988$$ 1.00000 0.0318142
$$989$$ −18.0000 −0.572367
$$990$$ 0 0
$$991$$ 44.0000 1.39771 0.698853 0.715265i $$-0.253694\pi$$
0.698853 + 0.715265i $$0.253694\pi$$
$$992$$ −8.00000 −0.254000
$$993$$ 0 0
$$994$$ 0 0
$$995$$ 16.0000 0.507234
$$996$$ 0 0
$$997$$ 22.0000 0.696747 0.348373 0.937356i $$-0.386734\pi$$
0.348373 + 0.937356i $$0.386734\pi$$
$$998$$ −4.00000 −0.126618
$$999$$ 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 4410.2.a.bg.1.1 1
3.2 odd 2 490.2.a.d.1.1 1
7.3 odd 6 630.2.k.d.541.1 2
7.5 odd 6 630.2.k.d.361.1 2
7.6 odd 2 4410.2.a.x.1.1 1
12.11 even 2 3920.2.a.e.1.1 1
15.2 even 4 2450.2.c.e.99.1 2
15.8 even 4 2450.2.c.e.99.2 2
15.14 odd 2 2450.2.a.v.1.1 1
21.2 odd 6 490.2.e.g.361.1 2
21.5 even 6 70.2.e.d.11.1 2
21.11 odd 6 490.2.e.g.471.1 2
21.17 even 6 70.2.e.d.51.1 yes 2
21.20 even 2 490.2.a.a.1.1 1
84.47 odd 6 560.2.q.b.81.1 2
84.59 odd 6 560.2.q.b.401.1 2
84.83 odd 2 3920.2.a.bh.1.1 1
105.17 odd 12 350.2.j.d.149.2 4
105.38 odd 12 350.2.j.d.149.1 4
105.47 odd 12 350.2.j.d.249.1 4
105.59 even 6 350.2.e.b.51.1 2
105.62 odd 4 2450.2.c.q.99.1 2
105.68 odd 12 350.2.j.d.249.2 4
105.83 odd 4 2450.2.c.q.99.2 2
105.89 even 6 350.2.e.b.151.1 2
105.104 even 2 2450.2.a.bf.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
70.2.e.d.11.1 2 21.5 even 6
70.2.e.d.51.1 yes 2 21.17 even 6
350.2.e.b.51.1 2 105.59 even 6
350.2.e.b.151.1 2 105.89 even 6
350.2.j.d.149.1 4 105.38 odd 12
350.2.j.d.149.2 4 105.17 odd 12
350.2.j.d.249.1 4 105.47 odd 12
350.2.j.d.249.2 4 105.68 odd 12
490.2.a.a.1.1 1 21.20 even 2
490.2.a.d.1.1 1 3.2 odd 2
490.2.e.g.361.1 2 21.2 odd 6
490.2.e.g.471.1 2 21.11 odd 6
560.2.q.b.81.1 2 84.47 odd 6
560.2.q.b.401.1 2 84.59 odd 6
630.2.k.d.361.1 2 7.5 odd 6
630.2.k.d.541.1 2 7.3 odd 6
2450.2.a.v.1.1 1 15.14 odd 2
2450.2.a.bf.1.1 1 105.104 even 2
2450.2.c.e.99.1 2 15.2 even 4
2450.2.c.e.99.2 2 15.8 even 4
2450.2.c.q.99.1 2 105.62 odd 4
2450.2.c.q.99.2 2 105.83 odd 4
3920.2.a.e.1.1 1 12.11 even 2
3920.2.a.bh.1.1 1 84.83 odd 2
4410.2.a.x.1.1 1 7.6 odd 2
4410.2.a.bg.1.1 1 1.1 even 1 trivial