# Properties

 Label 1050.2.a.i.1.1 Level $1050$ Weight $2$ Character 1050.1 Self dual yes Analytic conductor $8.384$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$1050 = 2 \cdot 3 \cdot 5^{2} \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 1050.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$8.38429221223$$ Analytic rank: $$1$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 42) Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 1050.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} -4.00000 q^{11} +1.00000 q^{12} -6.00000 q^{13} -1.00000 q^{14} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} -4.00000 q^{19} +1.00000 q^{21} +4.00000 q^{22} -8.00000 q^{23} -1.00000 q^{24} +6.00000 q^{26} +1.00000 q^{27} +1.00000 q^{28} -2.00000 q^{29} -1.00000 q^{32} -4.00000 q^{33} +2.00000 q^{34} +1.00000 q^{36} +10.0000 q^{37} +4.00000 q^{38} -6.00000 q^{39} -6.00000 q^{41} -1.00000 q^{42} +4.00000 q^{43} -4.00000 q^{44} +8.00000 q^{46} +1.00000 q^{48} +1.00000 q^{49} -2.00000 q^{51} -6.00000 q^{52} -6.00000 q^{53} -1.00000 q^{54} -1.00000 q^{56} -4.00000 q^{57} +2.00000 q^{58} +4.00000 q^{59} +6.00000 q^{61} +1.00000 q^{63} +1.00000 q^{64} +4.00000 q^{66} -4.00000 q^{67} -2.00000 q^{68} -8.00000 q^{69} +8.00000 q^{71} -1.00000 q^{72} -10.0000 q^{73} -10.0000 q^{74} -4.00000 q^{76} -4.00000 q^{77} +6.00000 q^{78} +1.00000 q^{81} +6.00000 q^{82} +4.00000 q^{83} +1.00000 q^{84} -4.00000 q^{86} -2.00000 q^{87} +4.00000 q^{88} -6.00000 q^{89} -6.00000 q^{91} -8.00000 q^{92} -1.00000 q^{96} +14.0000 q^{97} -1.00000 q^{98} -4.00000 q^{99} +O(q^{100})$$

## 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
$$7$$ 1.00000 0.377964
$$8$$ −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 $$-0.812833\pi$$
−0.832050 + 0.554700i $$0.812833\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 0 0
$$16$$ 1.00000 0.250000
$$17$$ −2.00000 −0.485071 −0.242536 0.970143i $$-0.577979\pi$$
−0.242536 + 0.970143i $$0.577979\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ 0 0
$$21$$ 1.00000 0.218218
$$22$$ 4.00000 0.852803
$$23$$ −8.00000 −1.66812 −0.834058 0.551677i $$-0.813988\pi$$
−0.834058 + 0.551677i $$0.813988\pi$$
$$24$$ −1.00000 −0.204124
$$25$$ 0 0
$$26$$ 6.00000 1.17670
$$27$$ 1.00000 0.192450
$$28$$ 1.00000 0.188982
$$29$$ −2.00000 −0.371391 −0.185695 0.982607i $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 0 0
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −4.00000 −0.696311
$$34$$ 2.00000 0.342997
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 10.0000 1.64399 0.821995 0.569495i $$-0.192861\pi$$
0.821995 + 0.569495i $$0.192861\pi$$
$$38$$ 4.00000 0.648886
$$39$$ −6.00000 −0.960769
$$40$$ 0 0
$$41$$ −6.00000 −0.937043 −0.468521 0.883452i $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ −1.00000 −0.154303
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ 0 0
$$46$$ 8.00000 1.17954
$$47$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$48$$ 1.00000 0.144338
$$49$$ 1.00000 0.142857
$$50$$ 0 0
$$51$$ −2.00000 −0.280056
$$52$$ −6.00000 −0.832050
$$53$$ −6.00000 −0.824163 −0.412082 0.911147i $$-0.635198\pi$$
−0.412082 + 0.911147i $$0.635198\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 0 0
$$56$$ −1.00000 −0.133631
$$57$$ −4.00000 −0.529813
$$58$$ 2.00000 0.262613
$$59$$ 4.00000 0.520756 0.260378 0.965507i $$-0.416153\pi$$
0.260378 + 0.965507i $$0.416153\pi$$
$$60$$ 0 0
$$61$$ 6.00000 0.768221 0.384111 0.923287i $$-0.374508\pi$$
0.384111 + 0.923287i $$0.374508\pi$$
$$62$$ 0 0
$$63$$ 1.00000 0.125988
$$64$$ 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 $$0.578571\pi$$
$$68$$ −2.00000 −0.242536
$$69$$ −8.00000 −0.963087
$$70$$ 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
$$73$$ −10.0000 −1.17041 −0.585206 0.810885i $$-0.698986\pi$$
−0.585206 + 0.810885i $$0.698986\pi$$
$$74$$ −10.0000 −1.16248
$$75$$ 0 0
$$76$$ −4.00000 −0.458831
$$77$$ −4.00000 −0.455842
$$78$$ 6.00000 0.679366
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 6.00000 0.662589
$$83$$ 4.00000 0.439057 0.219529 0.975606i $$-0.429548\pi$$
0.219529 + 0.975606i $$0.429548\pi$$
$$84$$ 1.00000 0.109109
$$85$$ 0 0
$$86$$ −4.00000 −0.431331
$$87$$ −2.00000 −0.214423
$$88$$ 4.00000 0.426401
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 0 0
$$91$$ −6.00000 −0.628971
$$92$$ −8.00000 −0.834058
$$93$$ 0 0
$$94$$ 0 0
$$95$$ 0 0
$$96$$ −1.00000 −0.102062
$$97$$ 14.0000 1.42148 0.710742 0.703452i $$-0.248359\pi$$
0.710742 + 0.703452i $$0.248359\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ −4.00000 −0.402015
$$100$$ 0 0
$$101$$ −2.00000 −0.199007 −0.0995037 0.995037i $$-0.531726\pi$$
−0.0995037 + 0.995037i $$0.531726\pi$$
$$102$$ 2.00000 0.198030
$$103$$ −8.00000 −0.788263 −0.394132 0.919054i $$-0.628955\pi$$
−0.394132 + 0.919054i $$0.628955\pi$$
$$104$$ 6.00000 0.588348
$$105$$ 0 0
$$106$$ 6.00000 0.582772
$$107$$ −12.0000 −1.16008 −0.580042 0.814587i $$-0.696964\pi$$
−0.580042 + 0.814587i $$0.696964\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 0 0
$$111$$ 10.0000 0.949158
$$112$$ 1.00000 0.0944911
$$113$$ 14.0000 1.31701 0.658505 0.752577i $$-0.271189\pi$$
0.658505 + 0.752577i $$0.271189\pi$$
$$114$$ 4.00000 0.374634
$$115$$ 0 0
$$116$$ −2.00000 −0.185695
$$117$$ −6.00000 −0.554700
$$118$$ −4.00000 −0.368230
$$119$$ −2.00000 −0.183340
$$120$$ 0 0
$$121$$ 5.00000 0.454545
$$122$$ −6.00000 −0.543214
$$123$$ −6.00000 −0.541002
$$124$$ 0 0
$$125$$ 0 0
$$126$$ −1.00000 −0.0890871
$$127$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 4.00000 0.352180
$$130$$ 0 0
$$131$$ −20.0000 −1.74741 −0.873704 0.486458i $$-0.838289\pi$$
−0.873704 + 0.486458i $$0.838289\pi$$
$$132$$ −4.00000 −0.348155
$$133$$ −4.00000 −0.346844
$$134$$ 4.00000 0.345547
$$135$$ 0 0
$$136$$ 2.00000 0.171499
$$137$$ −10.0000 −0.854358 −0.427179 0.904167i $$-0.640493\pi$$
−0.427179 + 0.904167i $$0.640493\pi$$
$$138$$ 8.00000 0.681005
$$139$$ 4.00000 0.339276 0.169638 0.985506i $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −8.00000 −0.671345
$$143$$ 24.0000 2.00698
$$144$$ 1.00000 0.0833333
$$145$$ 0 0
$$146$$ 10.0000 0.827606
$$147$$ 1.00000 0.0824786
$$148$$ 10.0000 0.821995
$$149$$ 6.00000 0.491539 0.245770 0.969328i $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 0 0
$$151$$ −8.00000 −0.651031 −0.325515 0.945537i $$-0.605538\pi$$
−0.325515 + 0.945537i $$0.605538\pi$$
$$152$$ 4.00000 0.324443
$$153$$ −2.00000 −0.161690
$$154$$ 4.00000 0.322329
$$155$$ 0 0
$$156$$ −6.00000 −0.480384
$$157$$ 10.0000 0.798087 0.399043 0.916932i $$-0.369342\pi$$
0.399043 + 0.916932i $$0.369342\pi$$
$$158$$ 0 0
$$159$$ −6.00000 −0.475831
$$160$$ 0 0
$$161$$ −8.00000 −0.630488
$$162$$ −1.00000 −0.0785674
$$163$$ −20.0000 −1.56652 −0.783260 0.621694i $$-0.786445\pi$$
−0.783260 + 0.621694i $$0.786445\pi$$
$$164$$ −6.00000 −0.468521
$$165$$ 0 0
$$166$$ −4.00000 −0.310460
$$167$$ 8.00000 0.619059 0.309529 0.950890i $$-0.399829\pi$$
0.309529 + 0.950890i $$0.399829\pi$$
$$168$$ −1.00000 −0.0771517
$$169$$ 23.0000 1.76923
$$170$$ 0 0
$$171$$ −4.00000 −0.305888
$$172$$ 4.00000 0.304997
$$173$$ −22.0000 −1.67263 −0.836315 0.548250i $$-0.815294\pi$$
−0.836315 + 0.548250i $$0.815294\pi$$
$$174$$ 2.00000 0.151620
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ 4.00000 0.300658
$$178$$ 6.00000 0.449719
$$179$$ −12.0000 −0.896922 −0.448461 0.893802i $$-0.648028\pi$$
−0.448461 + 0.893802i $$0.648028\pi$$
$$180$$ 0 0
$$181$$ −18.0000 −1.33793 −0.668965 0.743294i $$-0.733262\pi$$
−0.668965 + 0.743294i $$0.733262\pi$$
$$182$$ 6.00000 0.444750
$$183$$ 6.00000 0.443533
$$184$$ 8.00000 0.589768
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 8.00000 0.585018
$$188$$ 0 0
$$189$$ 1.00000 0.0727393
$$190$$ 0 0
$$191$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ −2.00000 −0.143963 −0.0719816 0.997406i $$-0.522932\pi$$
−0.0719816 + 0.997406i $$0.522932\pi$$
$$194$$ −14.0000 −1.00514
$$195$$ 0 0
$$196$$ 1.00000 0.0714286
$$197$$ 10.0000 0.712470 0.356235 0.934396i $$-0.384060\pi$$
0.356235 + 0.934396i $$0.384060\pi$$
$$198$$ 4.00000 0.284268
$$199$$ 8.00000 0.567105 0.283552 0.958957i $$-0.408487\pi$$
0.283552 + 0.958957i $$0.408487\pi$$
$$200$$ 0 0
$$201$$ −4.00000 −0.282138
$$202$$ 2.00000 0.140720
$$203$$ −2.00000 −0.140372
$$204$$ −2.00000 −0.140028
$$205$$ 0 0
$$206$$ 8.00000 0.557386
$$207$$ −8.00000 −0.556038
$$208$$ −6.00000 −0.416025
$$209$$ 16.0000 1.10674
$$210$$ 0 0
$$211$$ 20.0000 1.37686 0.688428 0.725304i $$-0.258301\pi$$
0.688428 + 0.725304i $$0.258301\pi$$
$$212$$ −6.00000 −0.412082
$$213$$ 8.00000 0.548151
$$214$$ 12.0000 0.820303
$$215$$ 0 0
$$216$$ −1.00000 −0.0680414
$$217$$ 0 0
$$218$$ 2.00000 0.135457
$$219$$ −10.0000 −0.675737
$$220$$ 0 0
$$221$$ 12.0000 0.807207
$$222$$ −10.0000 −0.671156
$$223$$ 16.0000 1.07144 0.535720 0.844396i $$-0.320040\pi$$
0.535720 + 0.844396i $$0.320040\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ 0 0
$$226$$ −14.0000 −0.931266
$$227$$ −12.0000 −0.796468 −0.398234 0.917284i $$-0.630377\pi$$
−0.398234 + 0.917284i $$0.630377\pi$$
$$228$$ −4.00000 −0.264906
$$229$$ −2.00000 −0.132164 −0.0660819 0.997814i $$-0.521050\pi$$
−0.0660819 + 0.997814i $$0.521050\pi$$
$$230$$ 0 0
$$231$$ −4.00000 −0.263181
$$232$$ 2.00000 0.131306
$$233$$ 22.0000 1.44127 0.720634 0.693316i $$-0.243851\pi$$
0.720634 + 0.693316i $$0.243851\pi$$
$$234$$ 6.00000 0.392232
$$235$$ 0 0
$$236$$ 4.00000 0.260378
$$237$$ 0 0
$$238$$ 2.00000 0.129641
$$239$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$240$$ 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
$$244$$ 6.00000 0.384111
$$245$$ 0 0
$$246$$ 6.00000 0.382546
$$247$$ 24.0000 1.52708
$$248$$ 0 0
$$249$$ 4.00000 0.253490
$$250$$ 0 0
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 1.00000 0.0629941
$$253$$ 32.0000 2.01182
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 1.00000 0.0625000
$$257$$ 30.0000 1.87135 0.935674 0.352865i $$-0.114792\pi$$
0.935674 + 0.352865i $$0.114792\pi$$
$$258$$ −4.00000 −0.249029
$$259$$ 10.0000 0.621370
$$260$$ 0 0
$$261$$ −2.00000 −0.123797
$$262$$ 20.0000 1.23560
$$263$$ 24.0000 1.47990 0.739952 0.672660i $$-0.234848\pi$$
0.739952 + 0.672660i $$0.234848\pi$$
$$264$$ 4.00000 0.246183
$$265$$ 0 0
$$266$$ 4.00000 0.245256
$$267$$ −6.00000 −0.367194
$$268$$ −4.00000 −0.244339
$$269$$ 22.0000 1.34136 0.670682 0.741745i $$-0.266002\pi$$
0.670682 + 0.741745i $$0.266002\pi$$
$$270$$ 0 0
$$271$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$272$$ −2.00000 −0.121268
$$273$$ −6.00000 −0.363137
$$274$$ 10.0000 0.604122
$$275$$ 0 0
$$276$$ −8.00000 −0.481543
$$277$$ 10.0000 0.600842 0.300421 0.953807i $$-0.402873\pi$$
0.300421 + 0.953807i $$0.402873\pi$$
$$278$$ −4.00000 −0.239904
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 26.0000 1.55103 0.775515 0.631329i $$-0.217490\pi$$
0.775515 + 0.631329i $$0.217490\pi$$
$$282$$ 0 0
$$283$$ −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
$$287$$ −6.00000 −0.354169
$$288$$ −1.00000 −0.0589256
$$289$$ −13.0000 −0.764706
$$290$$ 0 0
$$291$$ 14.0000 0.820695
$$292$$ −10.0000 −0.585206
$$293$$ −30.0000 −1.75262 −0.876309 0.481749i $$-0.840002\pi$$
−0.876309 + 0.481749i $$0.840002\pi$$
$$294$$ −1.00000 −0.0583212
$$295$$ 0 0
$$296$$ −10.0000 −0.581238
$$297$$ −4.00000 −0.232104
$$298$$ −6.00000 −0.347571
$$299$$ 48.0000 2.77591
$$300$$ 0 0
$$301$$ 4.00000 0.230556
$$302$$ 8.00000 0.460348
$$303$$ −2.00000 −0.114897
$$304$$ −4.00000 −0.229416
$$305$$ 0 0
$$306$$ 2.00000 0.114332
$$307$$ −28.0000 −1.59804 −0.799022 0.601302i $$-0.794649\pi$$
−0.799022 + 0.601302i $$0.794649\pi$$
$$308$$ −4.00000 −0.227921
$$309$$ −8.00000 −0.455104
$$310$$ 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 $$0.591202\pi$$
$$314$$ −10.0000 −0.564333
$$315$$ 0 0
$$316$$ 0 0
$$317$$ 18.0000 1.01098 0.505490 0.862832i $$-0.331312\pi$$
0.505490 + 0.862832i $$0.331312\pi$$
$$318$$ 6.00000 0.336463
$$319$$ 8.00000 0.447914
$$320$$ 0 0
$$321$$ −12.0000 −0.669775
$$322$$ 8.00000 0.445823
$$323$$ 8.00000 0.445132
$$324$$ 1.00000 0.0555556
$$325$$ 0 0
$$326$$ 20.0000 1.10770
$$327$$ −2.00000 −0.110600
$$328$$ 6.00000 0.331295
$$329$$ 0 0
$$330$$ 0 0
$$331$$ −4.00000 −0.219860 −0.109930 0.993939i $$-0.535063\pi$$
−0.109930 + 0.993939i $$0.535063\pi$$
$$332$$ 4.00000 0.219529
$$333$$ 10.0000 0.547997
$$334$$ −8.00000 −0.437741
$$335$$ 0 0
$$336$$ 1.00000 0.0545545
$$337$$ −18.0000 −0.980522 −0.490261 0.871576i $$-0.663099\pi$$
−0.490261 + 0.871576i $$0.663099\pi$$
$$338$$ −23.0000 −1.25104
$$339$$ 14.0000 0.760376
$$340$$ 0 0
$$341$$ 0 0
$$342$$ 4.00000 0.216295
$$343$$ 1.00000 0.0539949
$$344$$ −4.00000 −0.215666
$$345$$ 0 0
$$346$$ 22.0000 1.18273
$$347$$ −12.0000 −0.644194 −0.322097 0.946707i $$-0.604388\pi$$
−0.322097 + 0.946707i $$0.604388\pi$$
$$348$$ −2.00000 −0.107211
$$349$$ 22.0000 1.17763 0.588817 0.808267i $$-0.299594\pi$$
0.588817 + 0.808267i $$0.299594\pi$$
$$350$$ 0 0
$$351$$ −6.00000 −0.320256
$$352$$ 4.00000 0.213201
$$353$$ 30.0000 1.59674 0.798369 0.602168i $$-0.205696\pi$$
0.798369 + 0.602168i $$0.205696\pi$$
$$354$$ −4.00000 −0.212598
$$355$$ 0 0
$$356$$ −6.00000 −0.317999
$$357$$ −2.00000 −0.105851
$$358$$ 12.0000 0.634220
$$359$$ −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
$$362$$ 18.0000 0.946059
$$363$$ 5.00000 0.262432
$$364$$ −6.00000 −0.314485
$$365$$ 0 0
$$366$$ −6.00000 −0.313625
$$367$$ −32.0000 −1.67039 −0.835193 0.549957i $$-0.814644\pi$$
−0.835193 + 0.549957i $$0.814644\pi$$
$$368$$ −8.00000 −0.417029
$$369$$ −6.00000 −0.312348
$$370$$ 0 0
$$371$$ −6.00000 −0.311504
$$372$$ 0 0
$$373$$ −22.0000 −1.13912 −0.569558 0.821951i $$-0.692886\pi$$
−0.569558 + 0.821951i $$0.692886\pi$$
$$374$$ −8.00000 −0.413670
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 12.0000 0.618031
$$378$$ −1.00000 −0.0514344
$$379$$ −20.0000 −1.02733 −0.513665 0.857991i $$-0.671713\pi$$
−0.513665 + 0.857991i $$0.671713\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 0 0
$$383$$ 16.0000 0.817562 0.408781 0.912633i $$-0.365954\pi$$
0.408781 + 0.912633i $$0.365954\pi$$
$$384$$ −1.00000 −0.0510310
$$385$$ 0 0
$$386$$ 2.00000 0.101797
$$387$$ 4.00000 0.203331
$$388$$ 14.0000 0.710742
$$389$$ −26.0000 −1.31825 −0.659126 0.752032i $$-0.729074\pi$$
−0.659126 + 0.752032i $$0.729074\pi$$
$$390$$ 0 0
$$391$$ 16.0000 0.809155
$$392$$ −1.00000 −0.0505076
$$393$$ −20.0000 −1.00887
$$394$$ −10.0000 −0.503793
$$395$$ 0 0
$$396$$ −4.00000 −0.201008
$$397$$ −6.00000 −0.301131 −0.150566 0.988600i $$-0.548110\pi$$
−0.150566 + 0.988600i $$0.548110\pi$$
$$398$$ −8.00000 −0.401004
$$399$$ −4.00000 −0.200250
$$400$$ 0 0
$$401$$ 18.0000 0.898877 0.449439 0.893311i $$-0.351624\pi$$
0.449439 + 0.893311i $$0.351624\pi$$
$$402$$ 4.00000 0.199502
$$403$$ 0 0
$$404$$ −2.00000 −0.0995037
$$405$$ 0 0
$$406$$ 2.00000 0.0992583
$$407$$ −40.0000 −1.98273
$$408$$ 2.00000 0.0990148
$$409$$ −22.0000 −1.08783 −0.543915 0.839140i $$-0.683059\pi$$
−0.543915 + 0.839140i $$0.683059\pi$$
$$410$$ 0 0
$$411$$ −10.0000 −0.493264
$$412$$ −8.00000 −0.394132
$$413$$ 4.00000 0.196827
$$414$$ 8.00000 0.393179
$$415$$ 0 0
$$416$$ 6.00000 0.294174
$$417$$ 4.00000 0.195881
$$418$$ −16.0000 −0.782586
$$419$$ −36.0000 −1.75872 −0.879358 0.476162i $$-0.842028\pi$$
−0.879358 + 0.476162i $$0.842028\pi$$
$$420$$ 0 0
$$421$$ 6.00000 0.292422 0.146211 0.989253i $$-0.453292\pi$$
0.146211 + 0.989253i $$0.453292\pi$$
$$422$$ −20.0000 −0.973585
$$423$$ 0 0
$$424$$ 6.00000 0.291386
$$425$$ 0 0
$$426$$ −8.00000 −0.387601
$$427$$ 6.00000 0.290360
$$428$$ −12.0000 −0.580042
$$429$$ 24.0000 1.15873
$$430$$ 0 0
$$431$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$432$$ 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
$$437$$ 32.0000 1.53077
$$438$$ 10.0000 0.477818
$$439$$ −24.0000 −1.14546 −0.572729 0.819745i $$-0.694115\pi$$
−0.572729 + 0.819745i $$0.694115\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ −12.0000 −0.570782
$$443$$ 4.00000 0.190046 0.0950229 0.995475i $$-0.469708\pi$$
0.0950229 + 0.995475i $$0.469708\pi$$
$$444$$ 10.0000 0.474579
$$445$$ 0 0
$$446$$ −16.0000 −0.757622
$$447$$ 6.00000 0.283790
$$448$$ 1.00000 0.0472456
$$449$$ 34.0000 1.60456 0.802280 0.596948i $$-0.203620\pi$$
0.802280 + 0.596948i $$0.203620\pi$$
$$450$$ 0 0
$$451$$ 24.0000 1.13012
$$452$$ 14.0000 0.658505
$$453$$ −8.00000 −0.375873
$$454$$ 12.0000 0.563188
$$455$$ 0 0
$$456$$ 4.00000 0.187317
$$457$$ −10.0000 −0.467780 −0.233890 0.972263i $$-0.575146\pi$$
−0.233890 + 0.972263i $$0.575146\pi$$
$$458$$ 2.00000 0.0934539
$$459$$ −2.00000 −0.0933520
$$460$$ 0 0
$$461$$ 22.0000 1.02464 0.512321 0.858794i $$-0.328786\pi$$
0.512321 + 0.858794i $$0.328786\pi$$
$$462$$ 4.00000 0.186097
$$463$$ 32.0000 1.48717 0.743583 0.668644i $$-0.233125\pi$$
0.743583 + 0.668644i $$0.233125\pi$$
$$464$$ −2.00000 −0.0928477
$$465$$ 0 0
$$466$$ −22.0000 −1.01913
$$467$$ −28.0000 −1.29569 −0.647843 0.761774i $$-0.724329\pi$$
−0.647843 + 0.761774i $$0.724329\pi$$
$$468$$ −6.00000 −0.277350
$$469$$ −4.00000 −0.184703
$$470$$ 0 0
$$471$$ 10.0000 0.460776
$$472$$ −4.00000 −0.184115
$$473$$ −16.0000 −0.735681
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −2.00000 −0.0916698
$$477$$ −6.00000 −0.274721
$$478$$ 0 0
$$479$$ −16.0000 −0.731059 −0.365529 0.930800i $$-0.619112\pi$$
−0.365529 + 0.930800i $$0.619112\pi$$
$$480$$ 0 0
$$481$$ −60.0000 −2.73576
$$482$$ −2.00000 −0.0910975
$$483$$ −8.00000 −0.364013
$$484$$ 5.00000 0.227273
$$485$$ 0 0
$$486$$ −1.00000 −0.0453609
$$487$$ −8.00000 −0.362515 −0.181257 0.983436i $$-0.558017\pi$$
−0.181257 + 0.983436i $$0.558017\pi$$
$$488$$ −6.00000 −0.271607
$$489$$ −20.0000 −0.904431
$$490$$ 0 0
$$491$$ 12.0000 0.541552 0.270776 0.962642i $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ −6.00000 −0.270501
$$493$$ 4.00000 0.180151
$$494$$ −24.0000 −1.07981
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 8.00000 0.358849
$$498$$ −4.00000 −0.179244
$$499$$ −44.0000 −1.96971 −0.984855 0.173379i $$-0.944532\pi$$
−0.984855 + 0.173379i $$0.944532\pi$$
$$500$$ 0 0
$$501$$ 8.00000 0.357414
$$502$$ 12.0000 0.535586
$$503$$ −24.0000 −1.07011 −0.535054 0.844818i $$-0.679709\pi$$
−0.535054 + 0.844818i $$0.679709\pi$$
$$504$$ −1.00000 −0.0445435
$$505$$ 0 0
$$506$$ −32.0000 −1.42257
$$507$$ 23.0000 1.02147
$$508$$ 0 0
$$509$$ 6.00000 0.265945 0.132973 0.991120i $$-0.457548\pi$$
0.132973 + 0.991120i $$0.457548\pi$$
$$510$$ 0 0
$$511$$ −10.0000 −0.442374
$$512$$ −1.00000 −0.0441942
$$513$$ −4.00000 −0.176604
$$514$$ −30.0000 −1.32324
$$515$$ 0 0
$$516$$ 4.00000 0.176090
$$517$$ 0 0
$$518$$ −10.0000 −0.439375
$$519$$ −22.0000 −0.965693
$$520$$ 0 0
$$521$$ −6.00000 −0.262865 −0.131432 0.991325i $$-0.541958\pi$$
−0.131432 + 0.991325i $$0.541958\pi$$
$$522$$ 2.00000 0.0875376
$$523$$ −20.0000 −0.874539 −0.437269 0.899331i $$-0.644054\pi$$
−0.437269 + 0.899331i $$0.644054\pi$$
$$524$$ −20.0000 −0.873704
$$525$$ 0 0
$$526$$ −24.0000 −1.04645
$$527$$ 0 0
$$528$$ −4.00000 −0.174078
$$529$$ 41.0000 1.78261
$$530$$ 0 0
$$531$$ 4.00000 0.173585
$$532$$ −4.00000 −0.173422
$$533$$ 36.0000 1.55933
$$534$$ 6.00000 0.259645
$$535$$ 0 0
$$536$$ 4.00000 0.172774
$$537$$ −12.0000 −0.517838
$$538$$ −22.0000 −0.948487
$$539$$ −4.00000 −0.172292
$$540$$ 0 0
$$541$$ 30.0000 1.28980 0.644900 0.764267i $$-0.276899\pi$$
0.644900 + 0.764267i $$0.276899\pi$$
$$542$$ 0 0
$$543$$ −18.0000 −0.772454
$$544$$ 2.00000 0.0857493
$$545$$ 0 0
$$546$$ 6.00000 0.256776
$$547$$ 12.0000 0.513083 0.256541 0.966533i $$-0.417417\pi$$
0.256541 + 0.966533i $$0.417417\pi$$
$$548$$ −10.0000 −0.427179
$$549$$ 6.00000 0.256074
$$550$$ 0 0
$$551$$ 8.00000 0.340811
$$552$$ 8.00000 0.340503
$$553$$ 0 0
$$554$$ −10.0000 −0.424859
$$555$$ 0 0
$$556$$ 4.00000 0.169638
$$557$$ 2.00000 0.0847427 0.0423714 0.999102i $$-0.486509\pi$$
0.0423714 + 0.999102i $$0.486509\pi$$
$$558$$ 0 0
$$559$$ −24.0000 −1.01509
$$560$$ 0 0
$$561$$ 8.00000 0.337760
$$562$$ −26.0000 −1.09674
$$563$$ −44.0000 −1.85438 −0.927189 0.374593i $$-0.877783\pi$$
−0.927189 + 0.374593i $$0.877783\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 4.00000 0.168133
$$567$$ 1.00000 0.0419961
$$568$$ −8.00000 −0.335673
$$569$$ −6.00000 −0.251533 −0.125767 0.992060i $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ 0 0
$$571$$ 12.0000 0.502184 0.251092 0.967963i $$-0.419210\pi$$
0.251092 + 0.967963i $$0.419210\pi$$
$$572$$ 24.0000 1.00349
$$573$$ 0 0
$$574$$ 6.00000 0.250435
$$575$$ 0 0
$$576$$ 1.00000 0.0416667
$$577$$ −34.0000 −1.41544 −0.707719 0.706494i $$-0.750276\pi$$
−0.707719 + 0.706494i $$0.750276\pi$$
$$578$$ 13.0000 0.540729
$$579$$ −2.00000 −0.0831172
$$580$$ 0 0
$$581$$ 4.00000 0.165948
$$582$$ −14.0000 −0.580319
$$583$$ 24.0000 0.993978
$$584$$ 10.0000 0.413803
$$585$$ 0 0
$$586$$ 30.0000 1.23929
$$587$$ 28.0000 1.15568 0.577842 0.816149i $$-0.303895\pi$$
0.577842 + 0.816149i $$0.303895\pi$$
$$588$$ 1.00000 0.0412393
$$589$$ 0 0
$$590$$ 0 0
$$591$$ 10.0000 0.411345
$$592$$ 10.0000 0.410997
$$593$$ −18.0000 −0.739171 −0.369586 0.929197i $$-0.620500\pi$$
−0.369586 + 0.929197i $$0.620500\pi$$
$$594$$ 4.00000 0.164122
$$595$$ 0 0
$$596$$ 6.00000 0.245770
$$597$$ 8.00000 0.327418
$$598$$ −48.0000 −1.96287
$$599$$ 24.0000 0.980613 0.490307 0.871550i $$-0.336885\pi$$
0.490307 + 0.871550i $$0.336885\pi$$
$$600$$ 0 0
$$601$$ 26.0000 1.06056 0.530281 0.847822i $$-0.322086\pi$$
0.530281 + 0.847822i $$0.322086\pi$$
$$602$$ −4.00000 −0.163028
$$603$$ −4.00000 −0.162893
$$604$$ −8.00000 −0.325515
$$605$$ 0 0
$$606$$ 2.00000 0.0812444
$$607$$ −48.0000 −1.94826 −0.974130 0.225989i $$-0.927439\pi$$
−0.974130 + 0.225989i $$0.927439\pi$$
$$608$$ 4.00000 0.162221
$$609$$ −2.00000 −0.0810441
$$610$$ 0 0
$$611$$ 0 0
$$612$$ −2.00000 −0.0808452
$$613$$ 42.0000 1.69636 0.848182 0.529705i $$-0.177697\pi$$
0.848182 + 0.529705i $$0.177697\pi$$
$$614$$ 28.0000 1.12999
$$615$$ 0 0
$$616$$ 4.00000 0.161165
$$617$$ 22.0000 0.885687 0.442843 0.896599i $$-0.353970\pi$$
0.442843 + 0.896599i $$0.353970\pi$$
$$618$$ 8.00000 0.321807
$$619$$ −44.0000 −1.76851 −0.884255 0.467005i $$-0.845333\pi$$
−0.884255 + 0.467005i $$0.845333\pi$$
$$620$$ 0 0
$$621$$ −8.00000 −0.321029
$$622$$ 8.00000 0.320771
$$623$$ −6.00000 −0.240385
$$624$$ −6.00000 −0.240192
$$625$$ 0 0
$$626$$ 10.0000 0.399680
$$627$$ 16.0000 0.638978
$$628$$ 10.0000 0.399043
$$629$$ −20.0000 −0.797452
$$630$$ 0 0
$$631$$ 8.00000 0.318475 0.159237 0.987240i $$-0.449096\pi$$
0.159237 + 0.987240i $$0.449096\pi$$
$$632$$ 0 0
$$633$$ 20.0000 0.794929
$$634$$ −18.0000 −0.714871
$$635$$ 0 0
$$636$$ −6.00000 −0.237915
$$637$$ −6.00000 −0.237729
$$638$$ −8.00000 −0.316723
$$639$$ 8.00000 0.316475
$$640$$ 0 0
$$641$$ 2.00000 0.0789953 0.0394976 0.999220i $$-0.487424\pi$$
0.0394976 + 0.999220i $$0.487424\pi$$
$$642$$ 12.0000 0.473602
$$643$$ 4.00000 0.157745 0.0788723 0.996885i $$-0.474868\pi$$
0.0788723 + 0.996885i $$0.474868\pi$$
$$644$$ −8.00000 −0.315244
$$645$$ 0 0
$$646$$ −8.00000 −0.314756
$$647$$ −24.0000 −0.943537 −0.471769 0.881722i $$-0.656384\pi$$
−0.471769 + 0.881722i $$0.656384\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ −16.0000 −0.628055
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −20.0000 −0.783260
$$653$$ 18.0000 0.704394 0.352197 0.935926i $$-0.385435\pi$$
0.352197 + 0.935926i $$0.385435\pi$$
$$654$$ 2.00000 0.0782062
$$655$$ 0 0
$$656$$ −6.00000 −0.234261
$$657$$ −10.0000 −0.390137
$$658$$ 0 0
$$659$$ −28.0000 −1.09073 −0.545363 0.838200i $$-0.683608\pi$$
−0.545363 + 0.838200i $$0.683608\pi$$
$$660$$ 0 0
$$661$$ −2.00000 −0.0777910 −0.0388955 0.999243i $$-0.512384\pi$$
−0.0388955 + 0.999243i $$0.512384\pi$$
$$662$$ 4.00000 0.155464
$$663$$ 12.0000 0.466041
$$664$$ −4.00000 −0.155230
$$665$$ 0 0
$$666$$ −10.0000 −0.387492
$$667$$ 16.0000 0.619522
$$668$$ 8.00000 0.309529
$$669$$ 16.0000 0.618596
$$670$$ 0 0
$$671$$ −24.0000 −0.926510
$$672$$ −1.00000 −0.0385758
$$673$$ −2.00000 −0.0770943 −0.0385472 0.999257i $$-0.512273\pi$$
−0.0385472 + 0.999257i $$0.512273\pi$$
$$674$$ 18.0000 0.693334
$$675$$ 0 0
$$676$$ 23.0000 0.884615
$$677$$ 18.0000 0.691796 0.345898 0.938272i $$-0.387574\pi$$
0.345898 + 0.938272i $$0.387574\pi$$
$$678$$ −14.0000 −0.537667
$$679$$ 14.0000 0.537271
$$680$$ 0 0
$$681$$ −12.0000 −0.459841
$$682$$ 0 0
$$683$$ −12.0000 −0.459167 −0.229584 0.973289i $$-0.573736\pi$$
−0.229584 + 0.973289i $$0.573736\pi$$
$$684$$ −4.00000 −0.152944
$$685$$ 0 0
$$686$$ −1.00000 −0.0381802
$$687$$ −2.00000 −0.0763048
$$688$$ 4.00000 0.152499
$$689$$ 36.0000 1.37149
$$690$$ 0 0
$$691$$ −4.00000 −0.152167 −0.0760836 0.997101i $$-0.524242\pi$$
−0.0760836 + 0.997101i $$0.524242\pi$$
$$692$$ −22.0000 −0.836315
$$693$$ −4.00000 −0.151947
$$694$$ 12.0000 0.455514
$$695$$ 0 0
$$696$$ 2.00000 0.0758098
$$697$$ 12.0000 0.454532
$$698$$ −22.0000 −0.832712
$$699$$ 22.0000 0.832116
$$700$$ 0 0
$$701$$ −2.00000 −0.0755390 −0.0377695 0.999286i $$-0.512025\pi$$
−0.0377695 + 0.999286i $$0.512025\pi$$
$$702$$ 6.00000 0.226455
$$703$$ −40.0000 −1.50863
$$704$$ −4.00000 −0.150756
$$705$$ 0 0
$$706$$ −30.0000 −1.12906
$$707$$ −2.00000 −0.0752177
$$708$$ 4.00000 0.150329
$$709$$ −10.0000 −0.375558 −0.187779 0.982211i $$-0.560129\pi$$
−0.187779 + 0.982211i $$0.560129\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 6.00000 0.224860
$$713$$ 0 0
$$714$$ 2.00000 0.0748481
$$715$$ 0 0
$$716$$ −12.0000 −0.448461
$$717$$ 0 0
$$718$$ 8.00000 0.298557
$$719$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$720$$ 0 0
$$721$$ −8.00000 −0.297936
$$722$$ 3.00000 0.111648
$$723$$ 2.00000 0.0743808
$$724$$ −18.0000 −0.668965
$$725$$ 0 0
$$726$$ −5.00000 −0.185567
$$727$$ 8.00000 0.296704 0.148352 0.988935i $$-0.452603\pi$$
0.148352 + 0.988935i $$0.452603\pi$$
$$728$$ 6.00000 0.222375
$$729$$ 1.00000 0.0370370
$$730$$ 0 0
$$731$$ −8.00000 −0.295891
$$732$$ 6.00000 0.221766
$$733$$ −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
$$736$$ 8.00000 0.294884
$$737$$ 16.0000 0.589368
$$738$$ 6.00000 0.220863
$$739$$ −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
$$742$$ 6.00000 0.220267
$$743$$ −24.0000 −0.880475 −0.440237 0.897881i $$-0.645106\pi$$
−0.440237 + 0.897881i $$0.645106\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 22.0000 0.805477
$$747$$ 4.00000 0.146352
$$748$$ 8.00000 0.292509
$$749$$ −12.0000 −0.438470
$$750$$ 0 0
$$751$$ 48.0000 1.75154 0.875772 0.482724i $$-0.160353\pi$$
0.875772 + 0.482724i $$0.160353\pi$$
$$752$$ 0 0
$$753$$ −12.0000 −0.437304
$$754$$ −12.0000 −0.437014
$$755$$ 0 0
$$756$$ 1.00000 0.0363696
$$757$$ −6.00000 −0.218074 −0.109037 0.994038i $$-0.534777\pi$$
−0.109037 + 0.994038i $$0.534777\pi$$
$$758$$ 20.0000 0.726433
$$759$$ 32.0000 1.16153
$$760$$ 0 0
$$761$$ −22.0000 −0.797499 −0.398750 0.917060i $$-0.630556\pi$$
−0.398750 + 0.917060i $$0.630556\pi$$
$$762$$ 0 0
$$763$$ −2.00000 −0.0724049
$$764$$ 0 0
$$765$$ 0 0
$$766$$ −16.0000 −0.578103
$$767$$ −24.0000 −0.866590
$$768$$ 1.00000 0.0360844
$$769$$ −14.0000 −0.504853 −0.252426 0.967616i $$-0.581229\pi$$
−0.252426 + 0.967616i $$0.581229\pi$$
$$770$$ 0 0
$$771$$ 30.0000 1.08042
$$772$$ −2.00000 −0.0719816
$$773$$ 2.00000 0.0719350 0.0359675 0.999353i $$-0.488549\pi$$
0.0359675 + 0.999353i $$0.488549\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ 0 0
$$776$$ −14.0000 −0.502571
$$777$$ 10.0000 0.358748
$$778$$ 26.0000 0.932145
$$779$$ 24.0000 0.859889
$$780$$ 0 0
$$781$$ −32.0000 −1.14505
$$782$$ −16.0000 −0.572159
$$783$$ −2.00000 −0.0714742
$$784$$ 1.00000 0.0357143
$$785$$ 0 0
$$786$$ 20.0000 0.713376
$$787$$ 36.0000 1.28326 0.641631 0.767014i $$-0.278258\pi$$
0.641631 + 0.767014i $$0.278258\pi$$
$$788$$ 10.0000 0.356235
$$789$$ 24.0000 0.854423
$$790$$ 0 0
$$791$$ 14.0000 0.497783
$$792$$ 4.00000 0.142134
$$793$$ −36.0000 −1.27840
$$794$$ 6.00000 0.212932
$$795$$ 0 0
$$796$$ 8.00000 0.283552
$$797$$ −6.00000 −0.212531 −0.106265 0.994338i $$-0.533889\pi$$
−0.106265 + 0.994338i $$0.533889\pi$$
$$798$$ 4.00000 0.141598
$$799$$ 0 0
$$800$$ 0 0
$$801$$ −6.00000 −0.212000
$$802$$ −18.0000 −0.635602
$$803$$ 40.0000 1.41157
$$804$$ −4.00000 −0.141069
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 22.0000 0.774437
$$808$$ 2.00000 0.0703598
$$809$$ 10.0000 0.351581 0.175791 0.984428i $$-0.443752\pi$$
0.175791 + 0.984428i $$0.443752\pi$$
$$810$$ 0 0
$$811$$ −44.0000 −1.54505 −0.772524 0.634985i $$-0.781006\pi$$
−0.772524 + 0.634985i $$0.781006\pi$$
$$812$$ −2.00000 −0.0701862
$$813$$ 0 0
$$814$$ 40.0000 1.40200
$$815$$ 0 0
$$816$$ −2.00000 −0.0700140
$$817$$ −16.0000 −0.559769
$$818$$ 22.0000 0.769212
$$819$$ −6.00000 −0.209657
$$820$$ 0 0
$$821$$ 38.0000 1.32621 0.663105 0.748527i $$-0.269238\pi$$
0.663105 + 0.748527i $$0.269238\pi$$
$$822$$ 10.0000 0.348790
$$823$$ 56.0000 1.95204 0.976019 0.217687i $$-0.0698512\pi$$
0.976019 + 0.217687i $$0.0698512\pi$$
$$824$$ 8.00000 0.278693
$$825$$ 0 0
$$826$$ −4.00000 −0.139178
$$827$$ 36.0000 1.25184 0.625921 0.779886i $$-0.284723\pi$$
0.625921 + 0.779886i $$0.284723\pi$$
$$828$$ −8.00000 −0.278019
$$829$$ −26.0000 −0.903017 −0.451509 0.892267i $$-0.649114\pi$$
−0.451509 + 0.892267i $$0.649114\pi$$
$$830$$ 0 0
$$831$$ 10.0000 0.346896
$$832$$ −6.00000 −0.208013
$$833$$ −2.00000 −0.0692959
$$834$$ −4.00000 −0.138509
$$835$$ 0 0
$$836$$ 16.0000 0.553372
$$837$$ 0 0
$$838$$ 36.0000 1.24360
$$839$$ 56.0000 1.93333 0.966667 0.256036i $$-0.0824164\pi$$
0.966667 + 0.256036i $$0.0824164\pi$$
$$840$$ 0 0
$$841$$ −25.0000 −0.862069
$$842$$ −6.00000 −0.206774
$$843$$ 26.0000 0.895488
$$844$$ 20.0000 0.688428
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 5.00000 0.171802
$$848$$ −6.00000 −0.206041
$$849$$ −4.00000 −0.137280
$$850$$ 0 0
$$851$$ −80.0000 −2.74236
$$852$$ 8.00000 0.274075
$$853$$ −14.0000 −0.479351 −0.239675 0.970853i $$-0.577041\pi$$
−0.239675 + 0.970853i $$0.577041\pi$$
$$854$$ −6.00000 −0.205316
$$855$$ 0 0
$$856$$ 12.0000 0.410152
$$857$$ −42.0000 −1.43469 −0.717346 0.696717i $$-0.754643\pi$$
−0.717346 + 0.696717i $$0.754643\pi$$
$$858$$ −24.0000 −0.819346
$$859$$ 20.0000 0.682391 0.341196 0.939992i $$-0.389168\pi$$
0.341196 + 0.939992i $$0.389168\pi$$
$$860$$ 0 0
$$861$$ −6.00000 −0.204479
$$862$$ 0 0
$$863$$ 32.0000 1.08929 0.544646 0.838666i $$-0.316664\pi$$
0.544646 + 0.838666i $$0.316664\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ 0 0
$$866$$ 2.00000 0.0679628
$$867$$ −13.0000 −0.441503
$$868$$ 0 0
$$869$$ 0 0
$$870$$ 0 0
$$871$$ 24.0000 0.813209
$$872$$ 2.00000 0.0677285
$$873$$ 14.0000 0.473828
$$874$$ −32.0000 −1.08242
$$875$$ 0 0
$$876$$ −10.0000 −0.337869
$$877$$ 2.00000 0.0675352 0.0337676 0.999430i $$-0.489249\pi$$
0.0337676 + 0.999430i $$0.489249\pi$$
$$878$$ 24.0000 0.809961
$$879$$ −30.0000 −1.01187
$$880$$ 0 0
$$881$$ 18.0000 0.606435 0.303218 0.952921i $$-0.401939\pi$$
0.303218 + 0.952921i $$0.401939\pi$$
$$882$$ −1.00000 −0.0336718
$$883$$ −20.0000 −0.673054 −0.336527 0.941674i $$-0.609252\pi$$
−0.336527 + 0.941674i $$0.609252\pi$$
$$884$$ 12.0000 0.403604
$$885$$ 0 0
$$886$$ −4.00000 −0.134383
$$887$$ 24.0000 0.805841 0.402921 0.915235i $$-0.367995\pi$$
0.402921 + 0.915235i $$0.367995\pi$$
$$888$$ −10.0000 −0.335578
$$889$$ 0 0
$$890$$ 0 0
$$891$$ −4.00000 −0.134005
$$892$$ 16.0000 0.535720
$$893$$ 0 0
$$894$$ −6.00000 −0.200670
$$895$$ 0 0
$$896$$ −1.00000 −0.0334077
$$897$$ 48.0000 1.60267
$$898$$ −34.0000 −1.13459
$$899$$ 0 0
$$900$$ 0 0
$$901$$ 12.0000 0.399778
$$902$$ −24.0000 −0.799113
$$903$$ 4.00000 0.133112
$$904$$ −14.0000 −0.465633
$$905$$ 0 0
$$906$$ 8.00000 0.265782
$$907$$ −12.0000 −0.398453 −0.199227 0.979953i $$-0.563843\pi$$
−0.199227 + 0.979953i $$0.563843\pi$$
$$908$$ −12.0000 −0.398234
$$909$$ −2.00000 −0.0663358
$$910$$ 0 0
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ −4.00000 −0.132453
$$913$$ −16.0000 −0.529523
$$914$$ 10.0000 0.330771
$$915$$ 0 0
$$916$$ −2.00000 −0.0660819
$$917$$ −20.0000 −0.660458
$$918$$ 2.00000 0.0660098
$$919$$ 40.0000 1.31948 0.659739 0.751495i $$-0.270667\pi$$
0.659739 + 0.751495i $$0.270667\pi$$
$$920$$ 0 0
$$921$$ −28.0000 −0.922631
$$922$$ −22.0000 −0.724531
$$923$$ −48.0000 −1.57994
$$924$$ −4.00000 −0.131590
$$925$$ 0 0
$$926$$ −32.0000 −1.05159
$$927$$ −8.00000 −0.262754
$$928$$ 2.00000 0.0656532
$$929$$ 18.0000 0.590561 0.295280 0.955411i $$-0.404587\pi$$
0.295280 + 0.955411i $$0.404587\pi$$
$$930$$ 0 0
$$931$$ −4.00000 −0.131095
$$932$$ 22.0000 0.720634
$$933$$ −8.00000 −0.261908
$$934$$ 28.0000 0.916188
$$935$$ 0 0
$$936$$ 6.00000 0.196116
$$937$$ 22.0000 0.718709 0.359354 0.933201i $$-0.382997\pi$$
0.359354 + 0.933201i $$0.382997\pi$$
$$938$$ 4.00000 0.130605
$$939$$ −10.0000 −0.326338
$$940$$ 0 0
$$941$$ −26.0000 −0.847576 −0.423788 0.905761i $$-0.639300\pi$$
−0.423788 + 0.905761i $$0.639300\pi$$
$$942$$ −10.0000 −0.325818
$$943$$ 48.0000 1.56310
$$944$$ 4.00000 0.130189
$$945$$ 0 0
$$946$$ 16.0000 0.520205
$$947$$ −4.00000 −0.129983 −0.0649913 0.997886i $$-0.520702\pi$$
−0.0649913 + 0.997886i $$0.520702\pi$$
$$948$$ 0 0
$$949$$ 60.0000 1.94768
$$950$$ 0 0
$$951$$ 18.0000 0.583690
$$952$$ 2.00000 0.0648204
$$953$$ −26.0000 −0.842223 −0.421111 0.907009i $$-0.638360\pi$$
−0.421111 + 0.907009i $$0.638360\pi$$
$$954$$ 6.00000 0.194257
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 8.00000 0.258603
$$958$$ 16.0000 0.516937
$$959$$ −10.0000 −0.322917
$$960$$ 0 0
$$961$$ −31.0000 −1.00000
$$962$$ 60.0000 1.93448
$$963$$ −12.0000 −0.386695
$$964$$ 2.00000 0.0644157
$$965$$ 0 0
$$966$$ 8.00000 0.257396
$$967$$ −8.00000 −0.257263 −0.128631 0.991692i $$-0.541058\pi$$
−0.128631 + 0.991692i $$0.541058\pi$$
$$968$$ −5.00000 −0.160706
$$969$$ 8.00000 0.256997
$$970$$ 0 0
$$971$$ −12.0000 −0.385098 −0.192549 0.981287i $$-0.561675\pi$$
−0.192549 + 0.981287i $$0.561675\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 4.00000 0.128234
$$974$$ 8.00000 0.256337
$$975$$ 0 0
$$976$$ 6.00000 0.192055
$$977$$ −18.0000 −0.575871 −0.287936 0.957650i $$-0.592969\pi$$
−0.287936 + 0.957650i $$0.592969\pi$$
$$978$$ 20.0000 0.639529
$$979$$ 24.0000 0.767043
$$980$$ 0 0
$$981$$ −2.00000 −0.0638551
$$982$$ −12.0000 −0.382935
$$983$$ 24.0000 0.765481 0.382741 0.923856i $$-0.374980\pi$$
0.382741 + 0.923856i $$0.374980\pi$$
$$984$$ 6.00000 0.191273
$$985$$ 0 0
$$986$$ −4.00000 −0.127386
$$987$$ 0 0
$$988$$ 24.0000 0.763542
$$989$$ −32.0000 −1.01754
$$990$$ 0 0
$$991$$ −16.0000 −0.508257 −0.254128 0.967170i $$-0.581789\pi$$
−0.254128 + 0.967170i $$0.581789\pi$$
$$992$$ 0 0
$$993$$ −4.00000 −0.126936
$$994$$ −8.00000 −0.253745
$$995$$ 0 0
$$996$$ 4.00000 0.126745
$$997$$ −14.0000 −0.443384 −0.221692 0.975117i $$-0.571158\pi$$
−0.221692 + 0.975117i $$0.571158\pi$$
$$998$$ 44.0000 1.39280
$$999$$ 10.0000 0.316386
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 1050.2.a.i.1.1 1
3.2 odd 2 3150.2.a.bo.1.1 1
4.3 odd 2 8400.2.a.k.1.1 1
5.2 odd 4 1050.2.g.a.799.1 2
5.3 odd 4 1050.2.g.a.799.2 2
5.4 even 2 42.2.a.a.1.1 1
7.6 odd 2 7350.2.a.f.1.1 1
15.2 even 4 3150.2.g.r.2899.2 2
15.8 even 4 3150.2.g.r.2899.1 2
15.14 odd 2 126.2.a.a.1.1 1
20.19 odd 2 336.2.a.d.1.1 1
35.4 even 6 294.2.e.c.79.1 2
35.9 even 6 294.2.e.c.67.1 2
35.19 odd 6 294.2.e.a.67.1 2
35.24 odd 6 294.2.e.a.79.1 2
35.34 odd 2 294.2.a.g.1.1 1
40.19 odd 2 1344.2.a.i.1.1 1
40.29 even 2 1344.2.a.q.1.1 1
45.4 even 6 1134.2.f.g.379.1 2
45.14 odd 6 1134.2.f.j.379.1 2
45.29 odd 6 1134.2.f.j.757.1 2
45.34 even 6 1134.2.f.g.757.1 2
55.54 odd 2 5082.2.a.d.1.1 1
60.59 even 2 1008.2.a.j.1.1 1
65.64 even 2 7098.2.a.f.1.1 1
80.19 odd 4 5376.2.c.e.2689.2 2
80.29 even 4 5376.2.c.bc.2689.1 2
80.59 odd 4 5376.2.c.e.2689.1 2
80.69 even 4 5376.2.c.bc.2689.2 2
105.44 odd 6 882.2.g.h.361.1 2
105.59 even 6 882.2.g.j.667.1 2
105.74 odd 6 882.2.g.h.667.1 2
105.89 even 6 882.2.g.j.361.1 2
105.104 even 2 882.2.a.b.1.1 1
120.29 odd 2 4032.2.a.e.1.1 1
120.59 even 2 4032.2.a.m.1.1 1
140.19 even 6 2352.2.q.n.1537.1 2
140.39 odd 6 2352.2.q.i.961.1 2
140.59 even 6 2352.2.q.n.961.1 2
140.79 odd 6 2352.2.q.i.1537.1 2
140.139 even 2 2352.2.a.l.1.1 1
280.69 odd 2 9408.2.a.n.1.1 1
280.139 even 2 9408.2.a.bw.1.1 1
420.419 odd 2 7056.2.a.k.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
42.2.a.a.1.1 1 5.4 even 2
126.2.a.a.1.1 1 15.14 odd 2
294.2.a.g.1.1 1 35.34 odd 2
294.2.e.a.67.1 2 35.19 odd 6
294.2.e.a.79.1 2 35.24 odd 6
294.2.e.c.67.1 2 35.9 even 6
294.2.e.c.79.1 2 35.4 even 6
336.2.a.d.1.1 1 20.19 odd 2
882.2.a.b.1.1 1 105.104 even 2
882.2.g.h.361.1 2 105.44 odd 6
882.2.g.h.667.1 2 105.74 odd 6
882.2.g.j.361.1 2 105.89 even 6
882.2.g.j.667.1 2 105.59 even 6
1008.2.a.j.1.1 1 60.59 even 2
1050.2.a.i.1.1 1 1.1 even 1 trivial
1050.2.g.a.799.1 2 5.2 odd 4
1050.2.g.a.799.2 2 5.3 odd 4
1134.2.f.g.379.1 2 45.4 even 6
1134.2.f.g.757.1 2 45.34 even 6
1134.2.f.j.379.1 2 45.14 odd 6
1134.2.f.j.757.1 2 45.29 odd 6
1344.2.a.i.1.1 1 40.19 odd 2
1344.2.a.q.1.1 1 40.29 even 2
2352.2.a.l.1.1 1 140.139 even 2
2352.2.q.i.961.1 2 140.39 odd 6
2352.2.q.i.1537.1 2 140.79 odd 6
2352.2.q.n.961.1 2 140.59 even 6
2352.2.q.n.1537.1 2 140.19 even 6
3150.2.a.bo.1.1 1 3.2 odd 2
3150.2.g.r.2899.1 2 15.8 even 4
3150.2.g.r.2899.2 2 15.2 even 4
4032.2.a.e.1.1 1 120.29 odd 2
4032.2.a.m.1.1 1 120.59 even 2
5082.2.a.d.1.1 1 55.54 odd 2
5376.2.c.e.2689.1 2 80.59 odd 4
5376.2.c.e.2689.2 2 80.19 odd 4
5376.2.c.bc.2689.1 2 80.29 even 4
5376.2.c.bc.2689.2 2 80.69 even 4
7056.2.a.k.1.1 1 420.419 odd 2
7098.2.a.f.1.1 1 65.64 even 2
7350.2.a.f.1.1 1 7.6 odd 2
8400.2.a.k.1.1 1 4.3 odd 2
9408.2.a.n.1.1 1 280.69 odd 2
9408.2.a.bw.1.1 1 280.139 even 2