# Properties

 Label 5082.2.a.d.1.1 Level $5082$ Weight $2$ Character 5082.1 Self dual yes Analytic conductor $40.580$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$5082 = 2 \cdot 3 \cdot 7 \cdot 11^{2}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 5082.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$40.5799743072$$ 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 $$\chi$$ $$=$$ 5082.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} +1.00000 q^{6} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{9} +2.00000 q^{10} -1.00000 q^{12} -6.00000 q^{13} -1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} -2.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} -2.00000 q^{20} -1.00000 q^{21} +8.00000 q^{23} +1.00000 q^{24} -1.00000 q^{25} +6.00000 q^{26} -1.00000 q^{27} +1.00000 q^{28} +2.00000 q^{29} -2.00000 q^{30} -1.00000 q^{32} +2.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} -10.0000 q^{37} -4.00000 q^{38} +6.00000 q^{39} +2.00000 q^{40} +6.00000 q^{41} +1.00000 q^{42} +4.00000 q^{43} -2.00000 q^{45} -8.00000 q^{46} -1.00000 q^{48} +1.00000 q^{49} +1.00000 q^{50} +2.00000 q^{51} -6.00000 q^{52} +6.00000 q^{53} +1.00000 q^{54} -1.00000 q^{56} -4.00000 q^{57} -2.00000 q^{58} +4.00000 q^{59} +2.00000 q^{60} -6.00000 q^{61} +1.00000 q^{63} +1.00000 q^{64} +12.0000 q^{65} +4.00000 q^{67} -2.00000 q^{68} -8.00000 q^{69} +2.00000 q^{70} +8.00000 q^{71} -1.00000 q^{72} -10.0000 q^{73} +10.0000 q^{74} +1.00000 q^{75} +4.00000 q^{76} -6.00000 q^{78} -2.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +4.00000 q^{83} -1.00000 q^{84} +4.00000 q^{85} -4.00000 q^{86} -2.00000 q^{87} -6.00000 q^{89} +2.00000 q^{90} -6.00000 q^{91} +8.00000 q^{92} -8.00000 q^{95} +1.00000 q^{96} -14.0000 q^{97} -1.00000 q^{98} +O(q^{100})$$

## 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
$$3$$ −1.00000 −0.577350
$$4$$ 1.00000 0.500000
$$5$$ −2.00000 −0.894427 −0.447214 0.894427i $$-0.647584\pi$$
−0.447214 + 0.894427i $$0.647584\pi$$
$$6$$ 1.00000 0.408248
$$7$$ 1.00000 0.377964
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 2.00000 0.632456
$$11$$ 0 0
$$12$$ −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
$$15$$ 2.00000 0.516398
$$16$$ 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 $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ −2.00000 −0.447214
$$21$$ −1.00000 −0.218218
$$22$$ 0 0
$$23$$ 8.00000 1.66812 0.834058 0.551677i $$-0.186012\pi$$
0.834058 + 0.551677i $$0.186012\pi$$
$$24$$ 1.00000 0.204124
$$25$$ −1.00000 −0.200000
$$26$$ 6.00000 1.17670
$$27$$ −1.00000 −0.192450
$$28$$ 1.00000 0.188982
$$29$$ 2.00000 0.371391 0.185695 0.982607i $$-0.440546\pi$$
0.185695 + 0.982607i $$0.440546\pi$$
$$30$$ −2.00000 −0.365148
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ 0 0
$$34$$ 2.00000 0.342997
$$35$$ −2.00000 −0.338062
$$36$$ 1.00000 0.166667
$$37$$ −10.0000 −1.64399 −0.821995 0.569495i $$-0.807139\pi$$
−0.821995 + 0.569495i $$0.807139\pi$$
$$38$$ −4.00000 −0.648886
$$39$$ 6.00000 0.960769
$$40$$ 2.00000 0.316228
$$41$$ 6.00000 0.937043 0.468521 0.883452i $$-0.344787\pi$$
0.468521 + 0.883452i $$0.344787\pi$$
$$42$$ 1.00000 0.154303
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ 0 0
$$45$$ −2.00000 −0.298142
$$46$$ −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
$$50$$ 1.00000 0.141421
$$51$$ 2.00000 0.280056
$$52$$ −6.00000 −0.832050
$$53$$ 6.00000 0.824163 0.412082 0.911147i $$-0.364802\pi$$
0.412082 + 0.911147i $$0.364802\pi$$
$$54$$ 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 $$0.416153\pi$$
$$60$$ 2.00000 0.258199
$$61$$ −6.00000 −0.768221 −0.384111 0.923287i $$-0.625492\pi$$
−0.384111 + 0.923287i $$0.625492\pi$$
$$62$$ 0 0
$$63$$ 1.00000 0.125988
$$64$$ 1.00000 0.125000
$$65$$ 12.0000 1.48842
$$66$$ 0 0
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ −2.00000 −0.242536
$$69$$ −8.00000 −0.963087
$$70$$ 2.00000 0.239046
$$71$$ 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
$$75$$ 1.00000 0.115470
$$76$$ 4.00000 0.458831
$$77$$ 0 0
$$78$$ −6.00000 −0.679366
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ −2.00000 −0.223607
$$81$$ 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
$$85$$ 4.00000 0.433861
$$86$$ −4.00000 −0.431331
$$87$$ −2.00000 −0.214423
$$88$$ 0 0
$$89$$ −6.00000 −0.635999 −0.317999 0.948091i $$-0.603011\pi$$
−0.317999 + 0.948091i $$0.603011\pi$$
$$90$$ 2.00000 0.210819
$$91$$ −6.00000 −0.628971
$$92$$ 8.00000 0.834058
$$93$$ 0 0
$$94$$ 0 0
$$95$$ −8.00000 −0.820783
$$96$$ 1.00000 0.102062
$$97$$ −14.0000 −1.42148 −0.710742 0.703452i $$-0.751641\pi$$
−0.710742 + 0.703452i $$0.751641\pi$$
$$98$$ −1.00000 −0.101015
$$99$$ 0 0
$$100$$ −1.00000 −0.100000
$$101$$ 2.00000 0.199007 0.0995037 0.995037i $$-0.468274\pi$$
0.0995037 + 0.995037i $$0.468274\pi$$
$$102$$ −2.00000 −0.198030
$$103$$ 8.00000 0.788263 0.394132 0.919054i $$-0.371045\pi$$
0.394132 + 0.919054i $$0.371045\pi$$
$$104$$ 6.00000 0.588348
$$105$$ 2.00000 0.195180
$$106$$ −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 $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ 0 0
$$111$$ 10.0000 0.949158
$$112$$ 1.00000 0.0944911
$$113$$ −14.0000 −1.31701 −0.658505 0.752577i $$-0.728811\pi$$
−0.658505 + 0.752577i $$0.728811\pi$$
$$114$$ 4.00000 0.374634
$$115$$ −16.0000 −1.49201
$$116$$ 2.00000 0.185695
$$117$$ −6.00000 −0.554700
$$118$$ −4.00000 −0.368230
$$119$$ −2.00000 −0.183340
$$120$$ −2.00000 −0.182574
$$121$$ 0 0
$$122$$ 6.00000 0.543214
$$123$$ −6.00000 −0.541002
$$124$$ 0 0
$$125$$ 12.0000 1.07331
$$126$$ −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
$$130$$ −12.0000 −1.05247
$$131$$ 20.0000 1.74741 0.873704 0.486458i $$-0.161711\pi$$
0.873704 + 0.486458i $$0.161711\pi$$
$$132$$ 0 0
$$133$$ 4.00000 0.346844
$$134$$ −4.00000 −0.345547
$$135$$ 2.00000 0.172133
$$136$$ 2.00000 0.171499
$$137$$ 10.0000 0.854358 0.427179 0.904167i $$-0.359507\pi$$
0.427179 + 0.904167i $$0.359507\pi$$
$$138$$ 8.00000 0.681005
$$139$$ −4.00000 −0.339276 −0.169638 0.985506i $$-0.554260\pi$$
−0.169638 + 0.985506i $$0.554260\pi$$
$$140$$ −2.00000 −0.169031
$$141$$ 0 0
$$142$$ −8.00000 −0.671345
$$143$$ 0 0
$$144$$ 1.00000 0.0833333
$$145$$ −4.00000 −0.332182
$$146$$ 10.0000 0.827606
$$147$$ −1.00000 −0.0824786
$$148$$ −10.0000 −0.821995
$$149$$ −6.00000 −0.491539 −0.245770 0.969328i $$-0.579041\pi$$
−0.245770 + 0.969328i $$0.579041\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ 8.00000 0.651031 0.325515 0.945537i $$-0.394462\pi$$
0.325515 + 0.945537i $$0.394462\pi$$
$$152$$ −4.00000 −0.324443
$$153$$ −2.00000 −0.161690
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 6.00000 0.480384
$$157$$ −10.0000 −0.798087 −0.399043 0.916932i $$-0.630658\pi$$
−0.399043 + 0.916932i $$0.630658\pi$$
$$158$$ 0 0
$$159$$ −6.00000 −0.475831
$$160$$ 2.00000 0.158114
$$161$$ 8.00000 0.630488
$$162$$ −1.00000 −0.0785674
$$163$$ 20.0000 1.56652 0.783260 0.621694i $$-0.213555\pi$$
0.783260 + 0.621694i $$0.213555\pi$$
$$164$$ 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
$$170$$ −4.00000 −0.306786
$$171$$ 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
$$175$$ −1.00000 −0.0755929
$$176$$ 0 0
$$177$$ −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 $$0.648028\pi$$
$$180$$ −2.00000 −0.149071
$$181$$ −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
$$185$$ 20.0000 1.47043
$$186$$ 0 0
$$187$$ 0 0
$$188$$ 0 0
$$189$$ −1.00000 −0.0727393
$$190$$ 8.00000 0.580381
$$191$$ 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
$$195$$ −12.0000 −0.859338
$$196$$ 1.00000 0.0714286
$$197$$ 10.0000 0.712470 0.356235 0.934396i $$-0.384060\pi$$
0.356235 + 0.934396i $$0.384060\pi$$
$$198$$ 0 0
$$199$$ 8.00000 0.567105 0.283552 0.958957i $$-0.408487\pi$$
0.283552 + 0.958957i $$0.408487\pi$$
$$200$$ 1.00000 0.0707107
$$201$$ −4.00000 −0.282138
$$202$$ −2.00000 −0.140720
$$203$$ 2.00000 0.140372
$$204$$ 2.00000 0.140028
$$205$$ −12.0000 −0.838116
$$206$$ −8.00000 −0.557386
$$207$$ 8.00000 0.556038
$$208$$ −6.00000 −0.416025
$$209$$ 0 0
$$210$$ −2.00000 −0.138013
$$211$$ −20.0000 −1.37686 −0.688428 0.725304i $$-0.741699\pi$$
−0.688428 + 0.725304i $$0.741699\pi$$
$$212$$ 6.00000 0.412082
$$213$$ −8.00000 −0.548151
$$214$$ 12.0000 0.820303
$$215$$ −8.00000 −0.545595
$$216$$ 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 $$-0.679960\pi$$
−0.535720 + 0.844396i $$0.679960\pi$$
$$224$$ −1.00000 −0.0668153
$$225$$ −1.00000 −0.0666667
$$226$$ 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 $$0.521050\pi$$
$$230$$ 16.0000 1.05501
$$231$$ 0 0
$$232$$ −2.00000 −0.131306
$$233$$ 22.0000 1.44127 0.720634 0.693316i $$-0.243851\pi$$
0.720634 + 0.693316i $$0.243851\pi$$
$$234$$ 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 $$0.5\pi$$
$$240$$ 2.00000 0.129099
$$241$$ −2.00000 −0.128831 −0.0644157 0.997923i $$-0.520518\pi$$
−0.0644157 + 0.997923i $$0.520518\pi$$
$$242$$ 0 0
$$243$$ −1.00000 −0.0641500
$$244$$ −6.00000 −0.384111
$$245$$ −2.00000 −0.127775
$$246$$ 6.00000 0.382546
$$247$$ −24.0000 −1.52708
$$248$$ 0 0
$$249$$ −4.00000 −0.253490
$$250$$ −12.0000 −0.758947
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 1.00000 0.0629941
$$253$$ 0 0
$$254$$ 0 0
$$255$$ −4.00000 −0.250490
$$256$$ 1.00000 0.0625000
$$257$$ −30.0000 −1.87135 −0.935674 0.352865i $$-0.885208\pi$$
−0.935674 + 0.352865i $$0.885208\pi$$
$$258$$ 4.00000 0.249029
$$259$$ −10.0000 −0.621370
$$260$$ 12.0000 0.744208
$$261$$ 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 $$0.234848\pi$$
$$264$$ 0 0
$$265$$ −12.0000 −0.737154
$$266$$ −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 $$0.266002\pi$$
$$270$$ −2.00000 −0.121716
$$271$$ 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
$$280$$ 2.00000 0.119523
$$281$$ −26.0000 −1.55103 −0.775515 0.631329i $$-0.782510\pi$$
−0.775515 + 0.631329i $$0.782510\pi$$
$$282$$ 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
$$285$$ 8.00000 0.473879
$$286$$ 0 0
$$287$$ 6.00000 0.354169
$$288$$ −1.00000 −0.0589256
$$289$$ −13.0000 −0.764706
$$290$$ 4.00000 0.234888
$$291$$ 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
$$295$$ −8.00000 −0.465778
$$296$$ 10.0000 0.581238
$$297$$ 0 0
$$298$$ 6.00000 0.347571
$$299$$ −48.0000 −2.77591
$$300$$ 1.00000 0.0577350
$$301$$ 4.00000 0.230556
$$302$$ −8.00000 −0.460348
$$303$$ −2.00000 −0.114897
$$304$$ 4.00000 0.229416
$$305$$ 12.0000 0.687118
$$306$$ 2.00000 0.114332
$$307$$ −28.0000 −1.59804 −0.799022 0.601302i $$-0.794649\pi$$
−0.799022 + 0.601302i $$0.794649\pi$$
$$308$$ 0 0
$$309$$ −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 $$-0.408798\pi$$
0.282617 + 0.959233i $$0.408798\pi$$
$$314$$ 10.0000 0.564333
$$315$$ −2.00000 −0.112687
$$316$$ 0 0
$$317$$ −18.0000 −1.01098 −0.505490 0.862832i $$-0.668688\pi$$
−0.505490 + 0.862832i $$0.668688\pi$$
$$318$$ 6.00000 0.336463
$$319$$ 0 0
$$320$$ −2.00000 −0.111803
$$321$$ 12.0000 0.669775
$$322$$ −8.00000 −0.445823
$$323$$ −8.00000 −0.445132
$$324$$ 1.00000 0.0555556
$$325$$ 6.00000 0.332820
$$326$$ −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
$$335$$ −8.00000 −0.437087
$$336$$ −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
$$340$$ 4.00000 0.216930
$$341$$ 0 0
$$342$$ −4.00000 −0.216295
$$343$$ 1.00000 0.0539949
$$344$$ −4.00000 −0.215666
$$345$$ 16.0000 0.861411
$$346$$ 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 $$-0.700406\pi$$
−0.588817 + 0.808267i $$0.700406\pi$$
$$350$$ 1.00000 0.0534522
$$351$$ 6.00000 0.320256
$$352$$ 0 0
$$353$$ −30.0000 −1.59674 −0.798369 0.602168i $$-0.794304\pi$$
−0.798369 + 0.602168i $$0.794304\pi$$
$$354$$ 4.00000 0.212598
$$355$$ −16.0000 −0.849192
$$356$$ −6.00000 −0.317999
$$357$$ 2.00000 0.105851
$$358$$ 12.0000 0.634220
$$359$$ 8.00000 0.422224 0.211112 0.977462i $$-0.432292\pi$$
0.211112 + 0.977462i $$0.432292\pi$$
$$360$$ 2.00000 0.105409
$$361$$ −3.00000 −0.157895
$$362$$ 18.0000 0.946059
$$363$$ 0 0
$$364$$ −6.00000 −0.314485
$$365$$ 20.0000 1.04685
$$366$$ −6.00000 −0.313625
$$367$$ 32.0000 1.67039 0.835193 0.549957i $$-0.185356\pi$$
0.835193 + 0.549957i $$0.185356\pi$$
$$368$$ 8.00000 0.417029
$$369$$ 6.00000 0.312348
$$370$$ −20.0000 −1.03975
$$371$$ 6.00000 0.311504
$$372$$ 0 0
$$373$$ −22.0000 −1.13912 −0.569558 0.821951i $$-0.692886\pi$$
−0.569558 + 0.821951i $$0.692886\pi$$
$$374$$ 0 0
$$375$$ −12.0000 −0.619677
$$376$$ 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 $$0.671713\pi$$
$$380$$ −8.00000 −0.410391
$$381$$ 0 0
$$382$$ 0 0
$$383$$ −16.0000 −0.817562 −0.408781 0.912633i $$-0.634046\pi$$
−0.408781 + 0.912633i $$0.634046\pi$$
$$384$$ 1.00000 0.0510310
$$385$$ 0 0
$$386$$ 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 $$0.729074\pi$$
$$390$$ 12.0000 0.607644
$$391$$ −16.0000 −0.809155
$$392$$ −1.00000 −0.0505076
$$393$$ −20.0000 −1.00887
$$394$$ −10.0000 −0.503793
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 6.00000 0.301131 0.150566 0.988600i $$-0.451890\pi$$
0.150566 + 0.988600i $$0.451890\pi$$
$$398$$ −8.00000 −0.401004
$$399$$ −4.00000 −0.200250
$$400$$ −1.00000 −0.0500000
$$401$$ 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
$$405$$ −2.00000 −0.0993808
$$406$$ −2.00000 −0.0992583
$$407$$ 0 0
$$408$$ −2.00000 −0.0990148
$$409$$ 22.0000 1.08783 0.543915 0.839140i $$-0.316941\pi$$
0.543915 + 0.839140i $$0.316941\pi$$
$$410$$ 12.0000 0.592638
$$411$$ −10.0000 −0.493264
$$412$$ 8.00000 0.394132
$$413$$ 4.00000 0.196827
$$414$$ −8.00000 −0.393179
$$415$$ −8.00000 −0.392705
$$416$$ 6.00000 0.294174
$$417$$ 4.00000 0.195881
$$418$$ 0 0
$$419$$ −36.0000 −1.75872 −0.879358 0.476162i $$-0.842028\pi$$
−0.879358 + 0.476162i $$0.842028\pi$$
$$420$$ 2.00000 0.0975900
$$421$$ 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
$$425$$ 2.00000 0.0970143
$$426$$ 8.00000 0.387601
$$427$$ −6.00000 −0.290360
$$428$$ −12.0000 −0.580042
$$429$$ 0 0
$$430$$ 8.00000 0.385794
$$431$$ 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 $$-0.484697\pi$$
0.0480569 + 0.998845i $$0.484697\pi$$
$$434$$ 0 0
$$435$$ 4.00000 0.191785
$$436$$ 2.00000 0.0957826
$$437$$ 32.0000 1.53077
$$438$$ −10.0000 −0.477818
$$439$$ 24.0000 1.14546 0.572729 0.819745i $$-0.305885\pi$$
0.572729 + 0.819745i $$0.305885\pi$$
$$440$$ 0 0
$$441$$ 1.00000 0.0476190
$$442$$ −12.0000 −0.570782
$$443$$ −4.00000 −0.190046 −0.0950229 0.995475i $$-0.530292\pi$$
−0.0950229 + 0.995475i $$0.530292\pi$$
$$444$$ 10.0000 0.474579
$$445$$ 12.0000 0.568855
$$446$$ 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 $$0.203620\pi$$
$$450$$ 1.00000 0.0471405
$$451$$ 0 0
$$452$$ −14.0000 −0.658505
$$453$$ −8.00000 −0.375873
$$454$$ 12.0000 0.563188
$$455$$ 12.0000 0.562569
$$456$$ 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
$$460$$ −16.0000 −0.746004
$$461$$ −22.0000 −1.02464 −0.512321 0.858794i $$-0.671214\pi$$
−0.512321 + 0.858794i $$0.671214\pi$$
$$462$$ 0 0
$$463$$ −32.0000 −1.48717 −0.743583 0.668644i $$-0.766875\pi$$
−0.743583 + 0.668644i $$0.766875\pi$$
$$464$$ 2.00000 0.0928477
$$465$$ 0 0
$$466$$ −22.0000 −1.01913
$$467$$ 28.0000 1.29569 0.647843 0.761774i $$-0.275671\pi$$
0.647843 + 0.761774i $$0.275671\pi$$
$$468$$ −6.00000 −0.277350
$$469$$ 4.00000 0.184703
$$470$$ 0 0
$$471$$ 10.0000 0.460776
$$472$$ −4.00000 −0.184115
$$473$$ 0 0
$$474$$ 0 0
$$475$$ −4.00000 −0.183533
$$476$$ −2.00000 −0.0916698
$$477$$ 6.00000 0.274721
$$478$$ 0 0
$$479$$ 16.0000 0.731059 0.365529 0.930800i $$-0.380888\pi$$
0.365529 + 0.930800i $$0.380888\pi$$
$$480$$ −2.00000 −0.0912871
$$481$$ 60.0000 2.73576
$$482$$ 2.00000 0.0910975
$$483$$ −8.00000 −0.364013
$$484$$ 0 0
$$485$$ 28.0000 1.27141
$$486$$ 1.00000 0.0453609
$$487$$ 8.00000 0.362515 0.181257 0.983436i $$-0.441983\pi$$
0.181257 + 0.983436i $$0.441983\pi$$
$$488$$ 6.00000 0.271607
$$489$$ −20.0000 −0.904431
$$490$$ 2.00000 0.0903508
$$491$$ −12.0000 −0.541552 −0.270776 0.962642i $$-0.587280\pi$$
−0.270776 + 0.962642i $$0.587280\pi$$
$$492$$ −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 $$0.944532\pi$$
$$500$$ 12.0000 0.536656
$$501$$ −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
$$505$$ −4.00000 −0.177998
$$506$$ 0 0
$$507$$ −23.0000 −1.02147
$$508$$ 0 0
$$509$$ 6.00000 0.265945 0.132973 0.991120i $$-0.457548\pi$$
0.132973 + 0.991120i $$0.457548\pi$$
$$510$$ 4.00000 0.177123
$$511$$ −10.0000 −0.442374
$$512$$ −1.00000 −0.0441942
$$513$$ −4.00000 −0.176604
$$514$$ 30.0000 1.32324
$$515$$ −16.0000 −0.705044
$$516$$ −4.00000 −0.176090
$$517$$ 0 0
$$518$$ 10.0000 0.439375
$$519$$ 22.0000 0.965693
$$520$$ −12.0000 −0.526235
$$521$$ −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
$$525$$ 1.00000 0.0436436
$$526$$ −24.0000 −1.04645
$$527$$ 0 0
$$528$$ 0 0
$$529$$ 41.0000 1.78261
$$530$$ 12.0000 0.521247
$$531$$ 4.00000 0.173585
$$532$$ 4.00000 0.173422
$$533$$ −36.0000 −1.55933
$$534$$ −6.00000 −0.259645
$$535$$ 24.0000 1.03761
$$536$$ −4.00000 −0.172774
$$537$$ 12.0000 0.517838
$$538$$ −22.0000 −0.948487
$$539$$ 0 0
$$540$$ 2.00000 0.0860663
$$541$$ −30.0000 −1.28980 −0.644900 0.764267i $$-0.723101\pi$$
−0.644900 + 0.764267i $$0.723101\pi$$
$$542$$ 0 0
$$543$$ 18.0000 0.772454
$$544$$ 2.00000 0.0857493
$$545$$ −4.00000 −0.171341
$$546$$ −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
$$555$$ −20.0000 −0.848953
$$556$$ −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
$$560$$ −2.00000 −0.0845154
$$561$$ 0 0
$$562$$ 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
$$565$$ 28.0000 1.17797
$$566$$ 4.00000 0.168133
$$567$$ 1.00000 0.0419961
$$568$$ −8.00000 −0.335673
$$569$$ 6.00000 0.251533 0.125767 0.992060i $$-0.459861\pi$$
0.125767 + 0.992060i $$0.459861\pi$$
$$570$$ −8.00000 −0.335083
$$571$$ −12.0000 −0.502184 −0.251092 0.967963i $$-0.580790\pi$$
−0.251092 + 0.967963i $$0.580790\pi$$
$$572$$ 0 0
$$573$$ 0 0
$$574$$ −6.00000 −0.250435
$$575$$ −8.00000 −0.333623
$$576$$ 1.00000 0.0416667
$$577$$ 34.0000 1.41544 0.707719 0.706494i $$-0.249724\pi$$
0.707719 + 0.706494i $$0.249724\pi$$
$$578$$ 13.0000 0.540729
$$579$$ 2.00000 0.0831172
$$580$$ −4.00000 −0.166091
$$581$$ 4.00000 0.165948
$$582$$ −14.0000 −0.580319
$$583$$ 0 0
$$584$$ 10.0000 0.413803
$$585$$ 12.0000 0.496139
$$586$$ 30.0000 1.23929
$$587$$ −28.0000 −1.15568 −0.577842 0.816149i $$-0.696105\pi$$
−0.577842 + 0.816149i $$0.696105\pi$$
$$588$$ −1.00000 −0.0412393
$$589$$ 0 0
$$590$$ 8.00000 0.329355
$$591$$ −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 $$0.620500\pi$$
$$594$$ 0 0
$$595$$ 4.00000 0.163984
$$596$$ −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 $$0.336885\pi$$
$$600$$ −1.00000 −0.0408248
$$601$$ −26.0000 −1.06056 −0.530281 0.847822i $$-0.677914\pi$$
−0.530281 + 0.847822i $$0.677914\pi$$
$$602$$ −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
$$610$$ −12.0000 −0.485866
$$611$$ 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
$$615$$ 12.0000 0.483887
$$616$$ 0 0
$$617$$ −22.0000 −0.885687 −0.442843 0.896599i $$-0.646030\pi$$
−0.442843 + 0.896599i $$0.646030\pi$$
$$618$$ 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
$$625$$ −19.0000 −0.760000
$$626$$ −10.0000 −0.399680
$$627$$ 0 0
$$628$$ −10.0000 −0.399043
$$629$$ 20.0000 0.797452
$$630$$ 2.00000 0.0796819
$$631$$ 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
$$638$$ 0 0
$$639$$ 8.00000 0.316475
$$640$$ 2.00000 0.0790569
$$641$$ 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 $$-0.525132\pi$$
−0.0788723 + 0.996885i $$0.525132\pi$$
$$644$$ 8.00000 0.315244
$$645$$ 8.00000 0.315000
$$646$$ 8.00000 0.314756
$$647$$ 24.0000 0.943537 0.471769 0.881722i $$-0.343616\pi$$
0.471769 + 0.881722i $$0.343616\pi$$
$$648$$ −1.00000 −0.0392837
$$649$$ 0 0
$$650$$ −6.00000 −0.235339
$$651$$ 0 0
$$652$$ 20.0000 0.783260
$$653$$ −18.0000 −0.704394 −0.352197 0.935926i $$-0.614565\pi$$
−0.352197 + 0.935926i $$0.614565\pi$$
$$654$$ 2.00000 0.0782062
$$655$$ −40.0000 −1.56293
$$656$$ 6.00000 0.234261
$$657$$ −10.0000 −0.390137
$$658$$ 0 0
$$659$$ 28.0000 1.09073 0.545363 0.838200i $$-0.316392\pi$$
0.545363 + 0.838200i $$0.316392\pi$$
$$660$$ 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
$$665$$ −8.00000 −0.310227
$$666$$ 10.0000 0.387492
$$667$$ 16.0000 0.619522
$$668$$ 8.00000 0.309529
$$669$$ 16.0000 0.618596
$$670$$ 8.00000 0.309067
$$671$$ 0 0
$$672$$ 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
$$675$$ 1.00000 0.0384900
$$676$$ 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
$$680$$ −4.00000 −0.153393
$$681$$ 12.0000 0.459841
$$682$$ 0 0
$$683$$ 12.0000 0.459167 0.229584 0.973289i $$-0.426264\pi$$
0.229584 + 0.973289i $$0.426264\pi$$
$$684$$ 4.00000 0.152944
$$685$$ −20.0000 −0.764161
$$686$$ −1.00000 −0.0381802
$$687$$ 2.00000 0.0763048
$$688$$ 4.00000 0.152499
$$689$$ −36.0000 −1.37149
$$690$$ −16.0000 −0.609110
$$691$$ −4.00000 −0.152167 −0.0760836 0.997101i $$-0.524242\pi$$
−0.0760836 + 0.997101i $$0.524242\pi$$
$$692$$ −22.0000 −0.836315
$$693$$ 0 0
$$694$$ 12.0000 0.455514
$$695$$ 8.00000 0.303457
$$696$$ 2.00000 0.0758098
$$697$$ −12.0000 −0.454532
$$698$$ 22.0000 0.832712
$$699$$ −22.0000 −0.832116
$$700$$ −1.00000 −0.0377964
$$701$$ 2.00000 0.0755390 0.0377695 0.999286i $$-0.487975\pi$$
0.0377695 + 0.999286i $$0.487975\pi$$
$$702$$ −6.00000 −0.226455
$$703$$ −40.0000 −1.50863
$$704$$ 0 0
$$705$$ 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 $$0.560129\pi$$
$$710$$ 16.0000 0.600469
$$711$$ 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 $$0.5\pi$$
$$720$$ −2.00000 −0.0745356
$$721$$ 8.00000 0.297936
$$722$$ 3.00000 0.111648
$$723$$ 2.00000 0.0743808
$$724$$ −18.0000 −0.668965
$$725$$ −2.00000 −0.0742781
$$726$$ 0 0
$$727$$ −8.00000 −0.296704 −0.148352 0.988935i $$-0.547397\pi$$
−0.148352 + 0.988935i $$0.547397\pi$$
$$728$$ 6.00000 0.222375
$$729$$ 1.00000 0.0370370
$$730$$ −20.0000 −0.740233
$$731$$ −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
$$735$$ 2.00000 0.0737711
$$736$$ −8.00000 −0.294884
$$737$$ 0 0
$$738$$ −6.00000 −0.220863
$$739$$ 12.0000 0.441427 0.220714 0.975339i $$-0.429161\pi$$
0.220714 + 0.975339i $$0.429161\pi$$
$$740$$ 20.0000 0.735215
$$741$$ 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
$$745$$ 12.0000 0.439646
$$746$$ 22.0000 0.805477
$$747$$ 4.00000 0.146352
$$748$$ 0 0
$$749$$ −12.0000 −0.438470
$$750$$ 12.0000 0.438178
$$751$$ 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
$$755$$ −16.0000 −0.582300
$$756$$ −1.00000 −0.0363696
$$757$$ 6.00000 0.218074 0.109037 0.994038i $$-0.465223\pi$$
0.109037 + 0.994038i $$0.465223\pi$$
$$758$$ 20.0000 0.726433
$$759$$ 0 0
$$760$$ 8.00000 0.290191
$$761$$ 22.0000 0.797499 0.398750 0.917060i $$-0.369444\pi$$
0.398750 + 0.917060i $$0.369444\pi$$
$$762$$ 0 0
$$763$$ 2.00000 0.0724049
$$764$$ 0 0
$$765$$ 4.00000 0.144620
$$766$$ 16.0000 0.578103
$$767$$ −24.0000 −0.866590
$$768$$ −1.00000 −0.0360844
$$769$$ 14.0000 0.504853 0.252426 0.967616i $$-0.418771\pi$$
0.252426 + 0.967616i $$0.418771\pi$$
$$770$$ 0 0
$$771$$ 30.0000 1.08042
$$772$$ −2.00000 −0.0719816
$$773$$ −2.00000 −0.0719350 −0.0359675 0.999353i $$-0.511451\pi$$
−0.0359675 + 0.999353i $$0.511451\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ 0 0
$$776$$ 14.0000 0.502571
$$777$$ 10.0000 0.358748
$$778$$ 26.0000 0.932145
$$779$$ 24.0000 0.859889
$$780$$ −12.0000 −0.429669
$$781$$ 0 0
$$782$$ 16.0000 0.572159
$$783$$ −2.00000 −0.0714742
$$784$$ 1.00000 0.0357143
$$785$$ 20.0000 0.713831
$$786$$ 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
$$792$$ 0 0
$$793$$ 36.0000 1.27840
$$794$$ −6.00000 −0.212932
$$795$$ 12.0000 0.425596
$$796$$ 8.00000 0.283552
$$797$$ 6.00000 0.212531 0.106265 0.994338i $$-0.466111\pi$$
0.106265 + 0.994338i $$0.466111\pi$$
$$798$$ 4.00000 0.141598
$$799$$ 0 0
$$800$$ 1.00000 0.0353553
$$801$$ −6.00000 −0.212000
$$802$$ −18.0000 −0.635602
$$803$$ 0 0
$$804$$ −4.00000 −0.141069
$$805$$ −16.0000 −0.563926
$$806$$ 0 0
$$807$$ −22.0000 −0.774437
$$808$$ −2.00000 −0.0703598
$$809$$ −10.0000 −0.351581 −0.175791 0.984428i $$-0.556248\pi$$
−0.175791 + 0.984428i $$0.556248\pi$$
$$810$$ 2.00000 0.0702728
$$811$$ 44.0000 1.54505 0.772524 0.634985i $$-0.218994\pi$$
0.772524 + 0.634985i $$0.218994\pi$$
$$812$$ 2.00000 0.0701862
$$813$$ 0 0
$$814$$ 0 0
$$815$$ −40.0000 −1.40114
$$816$$ 2.00000 0.0700140
$$817$$ 16.0000 0.559769
$$818$$ −22.0000 −0.769212
$$819$$ −6.00000 −0.209657
$$820$$ −12.0000 −0.419058
$$821$$ −38.0000 −1.32621 −0.663105 0.748527i $$-0.730762\pi$$
−0.663105 + 0.748527i $$0.730762\pi$$
$$822$$ 10.0000 0.348790
$$823$$ −56.0000 −1.95204 −0.976019 0.217687i $$-0.930149\pi$$
−0.976019 + 0.217687i $$0.930149\pi$$
$$824$$ −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 $$0.649114\pi$$
$$830$$ 8.00000 0.277684
$$831$$ −10.0000 −0.346896
$$832$$ −6.00000 −0.208013
$$833$$ −2.00000 −0.0692959
$$834$$ −4.00000 −0.138509
$$835$$ −16.0000 −0.553703
$$836$$ 0 0
$$837$$ 0 0
$$838$$ 36.0000 1.24360
$$839$$ 56.0000 1.93333 0.966667 0.256036i $$-0.0824164\pi$$
0.966667 + 0.256036i $$0.0824164\pi$$
$$840$$ −2.00000 −0.0690066
$$841$$ −25.0000 −0.862069
$$842$$ −6.00000 −0.206774
$$843$$ 26.0000 0.895488
$$844$$ −20.0000 −0.688428
$$845$$ −46.0000 −1.58245
$$846$$ 0 0
$$847$$ 0 0
$$848$$ 6.00000 0.206041
$$849$$ 4.00000 0.137280
$$850$$ −2.00000 −0.0685994
$$851$$ −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
$$855$$ −8.00000 −0.273594
$$856$$ 12.0000 0.410152
$$857$$ −42.0000 −1.43469 −0.717346 0.696717i $$-0.754643\pi$$
−0.717346 + 0.696717i $$0.754643\pi$$
$$858$$ 0 0
$$859$$ 20.0000 0.682391 0.341196 0.939992i $$-0.389168\pi$$
0.341196 + 0.939992i $$0.389168\pi$$
$$860$$ −8.00000 −0.272798
$$861$$ −6.00000 −0.204479
$$862$$ 0 0
$$863$$ −32.0000 −1.08929 −0.544646 0.838666i $$-0.683336\pi$$
−0.544646 + 0.838666i $$0.683336\pi$$
$$864$$ 1.00000 0.0340207
$$865$$ 44.0000 1.49604
$$866$$ −2.00000 −0.0679628
$$867$$ 13.0000 0.441503
$$868$$ 0 0
$$869$$ 0 0
$$870$$ −4.00000 −0.135613
$$871$$ −24.0000 −0.813209
$$872$$ −2.00000 −0.0677285
$$873$$ −14.0000 −0.473828
$$874$$ −32.0000 −1.08242
$$875$$ 12.0000 0.405674
$$876$$ 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 $$-0.390748\pi$$
0.336527 + 0.941674i $$0.390748\pi$$
$$884$$ 12.0000 0.403604
$$885$$ 8.00000 0.268917
$$886$$ 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
$$890$$ −12.0000 −0.402241
$$891$$ 0 0
$$892$$ −16.0000 −0.535720
$$893$$ 0 0
$$894$$ −6.00000 −0.200670
$$895$$ 24.0000 0.802232
$$896$$ −1.00000 −0.0334077
$$897$$ 48.0000 1.60267
$$898$$ −34.0000 −1.13459
$$899$$ 0 0
$$900$$ −1.00000 −0.0333333
$$901$$ −12.0000 −0.399778
$$902$$ 0 0
$$903$$ −4.00000 −0.133112
$$904$$ 14.0000 0.465633
$$905$$ 36.0000 1.19668
$$906$$ 8.00000 0.265782
$$907$$ 12.0000 0.398453 0.199227 0.979953i $$-0.436157\pi$$
0.199227 + 0.979953i $$0.436157\pi$$
$$908$$ −12.0000 −0.398234
$$909$$ 2.00000 0.0663358
$$910$$ −12.0000 −0.397796
$$911$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$912$$ −4.00000 −0.132453
$$913$$ 0 0
$$914$$ 10.0000 0.330771
$$915$$ −12.0000 −0.396708
$$916$$ −2.00000 −0.0660819
$$917$$ 20.0000 0.660458
$$918$$ −2.00000 −0.0660098
$$919$$ −40.0000 −1.31948 −0.659739 0.751495i $$-0.729333\pi$$
−0.659739 + 0.751495i $$0.729333\pi$$
$$920$$ 16.0000 0.527504
$$921$$ 28.0000 0.922631
$$922$$ 22.0000 0.724531
$$923$$ −48.0000 −1.57994
$$924$$ 0 0
$$925$$ 10.0000 0.328798
$$926$$ 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 $$-0.360700\pi$$
0.423788 + 0.905761i $$0.360700\pi$$
$$942$$ −10.0000 −0.325818
$$943$$ 48.0000 1.56310
$$944$$ 4.00000 0.130189
$$945$$ 2.00000 0.0650600
$$946$$ 0 0
$$947$$ 4.00000 0.129983 0.0649913 0.997886i $$-0.479298\pi$$
0.0649913 + 0.997886i $$0.479298\pi$$
$$948$$ 0 0
$$949$$ 60.0000 1.94768
$$950$$ 4.00000 0.129777
$$951$$ 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
$$957$$ 0 0
$$958$$ −16.0000 −0.516937
$$959$$ 10.0000 0.322917
$$960$$ 2.00000 0.0645497
$$961$$ −31.0000 −1.00000
$$962$$ −60.0000 −1.93448
$$963$$ −12.0000 −0.386695
$$964$$ −2.00000 −0.0644157
$$965$$ 4.00000 0.128765
$$966$$ 8.00000 0.257396
$$967$$ −8.00000 −0.257263 −0.128631 0.991692i $$-0.541058\pi$$
−0.128631 + 0.991692i $$0.541058\pi$$
$$968$$ 0 0
$$969$$ 8.00000 0.256997
$$970$$ −28.0000 −0.899026
$$971$$ −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
$$975$$ −6.00000 −0.192154
$$976$$ −6.00000 −0.192055
$$977$$ 18.0000 0.575871 0.287936 0.957650i $$-0.407031\pi$$
0.287936 + 0.957650i $$0.407031\pi$$
$$978$$ 20.0000 0.639529
$$979$$ 0 0
$$980$$ −2.00000 −0.0638877
$$981$$ 2.00000 0.0638551
$$982$$ 12.0000 0.382935
$$983$$ −24.0000 −0.765481 −0.382741 0.923856i $$-0.625020\pi$$
−0.382741 + 0.923856i $$0.625020\pi$$
$$984$$ 6.00000 0.191273
$$985$$ −20.0000 −0.637253
$$986$$ 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
$$995$$ −16.0000 −0.507234
$$996$$ −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 5082.2.a.d.1.1 1
11.10 odd 2 42.2.a.a.1.1 1
33.32 even 2 126.2.a.a.1.1 1
44.43 even 2 336.2.a.d.1.1 1
55.32 even 4 1050.2.g.a.799.2 2
55.43 even 4 1050.2.g.a.799.1 2
55.54 odd 2 1050.2.a.i.1.1 1
77.10 even 6 294.2.e.a.79.1 2
77.32 odd 6 294.2.e.c.79.1 2
77.54 even 6 294.2.e.a.67.1 2
77.65 odd 6 294.2.e.c.67.1 2
77.76 even 2 294.2.a.g.1.1 1
88.21 odd 2 1344.2.a.q.1.1 1
88.43 even 2 1344.2.a.i.1.1 1
99.32 even 6 1134.2.f.j.379.1 2
99.43 odd 6 1134.2.f.g.757.1 2
99.65 even 6 1134.2.f.j.757.1 2
99.76 odd 6 1134.2.f.g.379.1 2
132.131 odd 2 1008.2.a.j.1.1 1
143.142 odd 2 7098.2.a.f.1.1 1
165.32 odd 4 3150.2.g.r.2899.1 2
165.98 odd 4 3150.2.g.r.2899.2 2
165.164 even 2 3150.2.a.bo.1.1 1
176.21 odd 4 5376.2.c.bc.2689.2 2
176.43 even 4 5376.2.c.e.2689.1 2
176.109 odd 4 5376.2.c.bc.2689.1 2
176.131 even 4 5376.2.c.e.2689.2 2
220.219 even 2 8400.2.a.k.1.1 1
231.32 even 6 882.2.g.h.667.1 2
231.65 even 6 882.2.g.h.361.1 2
231.131 odd 6 882.2.g.j.361.1 2
231.164 odd 6 882.2.g.j.667.1 2
231.230 odd 2 882.2.a.b.1.1 1
264.131 odd 2 4032.2.a.m.1.1 1
264.197 even 2 4032.2.a.e.1.1 1
308.87 odd 6 2352.2.q.n.961.1 2
308.131 odd 6 2352.2.q.n.1537.1 2
308.219 even 6 2352.2.q.i.1537.1 2
308.263 even 6 2352.2.q.i.961.1 2
308.307 odd 2 2352.2.a.l.1.1 1
385.384 even 2 7350.2.a.f.1.1 1
616.307 odd 2 9408.2.a.bw.1.1 1
616.461 even 2 9408.2.a.n.1.1 1
924.923 even 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 11.10 odd 2
126.2.a.a.1.1 1 33.32 even 2
294.2.a.g.1.1 1 77.76 even 2
294.2.e.a.67.1 2 77.54 even 6
294.2.e.a.79.1 2 77.10 even 6
294.2.e.c.67.1 2 77.65 odd 6
294.2.e.c.79.1 2 77.32 odd 6
336.2.a.d.1.1 1 44.43 even 2
882.2.a.b.1.1 1 231.230 odd 2
882.2.g.h.361.1 2 231.65 even 6
882.2.g.h.667.1 2 231.32 even 6
882.2.g.j.361.1 2 231.131 odd 6
882.2.g.j.667.1 2 231.164 odd 6
1008.2.a.j.1.1 1 132.131 odd 2
1050.2.a.i.1.1 1 55.54 odd 2
1050.2.g.a.799.1 2 55.43 even 4
1050.2.g.a.799.2 2 55.32 even 4
1134.2.f.g.379.1 2 99.76 odd 6
1134.2.f.g.757.1 2 99.43 odd 6
1134.2.f.j.379.1 2 99.32 even 6
1134.2.f.j.757.1 2 99.65 even 6
1344.2.a.i.1.1 1 88.43 even 2
1344.2.a.q.1.1 1 88.21 odd 2
2352.2.a.l.1.1 1 308.307 odd 2
2352.2.q.i.961.1 2 308.263 even 6
2352.2.q.i.1537.1 2 308.219 even 6
2352.2.q.n.961.1 2 308.87 odd 6
2352.2.q.n.1537.1 2 308.131 odd 6
3150.2.a.bo.1.1 1 165.164 even 2
3150.2.g.r.2899.1 2 165.32 odd 4
3150.2.g.r.2899.2 2 165.98 odd 4
4032.2.a.e.1.1 1 264.197 even 2
4032.2.a.m.1.1 1 264.131 odd 2
5082.2.a.d.1.1 1 1.1 even 1 trivial
5376.2.c.e.2689.1 2 176.43 even 4
5376.2.c.e.2689.2 2 176.131 even 4
5376.2.c.bc.2689.1 2 176.109 odd 4
5376.2.c.bc.2689.2 2 176.21 odd 4
7056.2.a.k.1.1 1 924.923 even 2
7098.2.a.f.1.1 1 143.142 odd 2
7350.2.a.f.1.1 1 385.384 even 2
8400.2.a.k.1.1 1 220.219 even 2
9408.2.a.n.1.1 1 616.461 even 2
9408.2.a.bw.1.1 1 616.307 odd 2