Properties

 Label 42.2.a.a.1.1 Level $42$ Weight $2$ Character 42.1 Self dual yes Analytic conductor $0.335$ Analytic rank $0$ Dimension $1$ CM no Inner twists $1$

Related objects

Newspace parameters

 Level: $$N$$ $$=$$ $$42 = 2 \cdot 3 \cdot 7$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 42.a (trivial)

Newform invariants

 Self dual: yes Analytic conductor: $$0.335371688489$$ Analytic rank: $$0$$ Dimension: $$1$$ Coefficient field: $$\mathbb{Q}$$ Coefficient ring: $$\mathbb{Z}$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

 Embedding label 1.1 Character $$\chi$$ $$=$$ 42.1

$q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -2.00000 q^{5} -1.00000 q^{6} -1.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -2.00000 q^{10} -4.00000 q^{11} -1.00000 q^{12} +6.00000 q^{13} -1.00000 q^{14} +2.00000 q^{15} +1.00000 q^{16} +2.00000 q^{17} +1.00000 q^{18} -4.00000 q^{19} -2.00000 q^{20} +1.00000 q^{21} -4.00000 q^{22} +8.00000 q^{23} -1.00000 q^{24} -1.00000 q^{25} +6.00000 q^{26} -1.00000 q^{27} -1.00000 q^{28} -2.00000 q^{29} +2.00000 q^{30} +1.00000 q^{32} +4.00000 q^{33} +2.00000 q^{34} +2.00000 q^{35} +1.00000 q^{36} -10.0000 q^{37} -4.00000 q^{38} -6.00000 q^{39} -2.00000 q^{40} -6.00000 q^{41} +1.00000 q^{42} -4.00000 q^{43} -4.00000 q^{44} -2.00000 q^{45} +8.00000 q^{46} -1.00000 q^{48} +1.00000 q^{49} -1.00000 q^{50} -2.00000 q^{51} +6.00000 q^{52} +6.00000 q^{53} -1.00000 q^{54} +8.00000 q^{55} -1.00000 q^{56} +4.00000 q^{57} -2.00000 q^{58} +4.00000 q^{59} +2.00000 q^{60} +6.00000 q^{61} -1.00000 q^{63} +1.00000 q^{64} -12.0000 q^{65} +4.00000 q^{66} +4.00000 q^{67} +2.00000 q^{68} -8.00000 q^{69} +2.00000 q^{70} +8.00000 q^{71} +1.00000 q^{72} +10.0000 q^{73} -10.0000 q^{74} +1.00000 q^{75} -4.00000 q^{76} +4.00000 q^{77} -6.00000 q^{78} -2.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} -4.00000 q^{83} +1.00000 q^{84} -4.00000 q^{85} -4.00000 q^{86} +2.00000 q^{87} -4.00000 q^{88} -6.00000 q^{89} -2.00000 q^{90} -6.00000 q^{91} +8.00000 q^{92} +8.00000 q^{95} -1.00000 q^{96} -14.0000 q^{97} +1.00000 q^{98} -4.00000 q^{99} +O(q^{100})$$

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
$$5$$ −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
$$11$$ −4.00000 −1.20605 −0.603023 0.797724i $$-0.706037\pi$$
−0.603023 + 0.797724i $$0.706037\pi$$
$$12$$ −1.00000 −0.288675
$$13$$ 6.00000 1.66410 0.832050 0.554700i $$-0.187167\pi$$
0.832050 + 0.554700i $$0.187167\pi$$
$$14$$ −1.00000 −0.267261
$$15$$ 2.00000 0.516398
$$16$$ 1.00000 0.250000
$$17$$ 2.00000 0.485071 0.242536 0.970143i $$-0.422021\pi$$
0.242536 + 0.970143i $$0.422021\pi$$
$$18$$ 1.00000 0.235702
$$19$$ −4.00000 −0.917663 −0.458831 0.888523i $$-0.651732\pi$$
−0.458831 + 0.888523i $$0.651732\pi$$
$$20$$ −2.00000 −0.447214
$$21$$ 1.00000 0.218218
$$22$$ −4.00000 −0.852803
$$23$$ 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 $$-0.559454\pi$$
−0.185695 + 0.982607i $$0.559454\pi$$
$$30$$ 2.00000 0.365148
$$31$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$32$$ 1.00000 0.176777
$$33$$ 4.00000 0.696311
$$34$$ 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 $$-0.655213\pi$$
−0.468521 + 0.883452i $$0.655213\pi$$
$$42$$ 1.00000 0.154303
$$43$$ −4.00000 −0.609994 −0.304997 0.952353i $$-0.598656\pi$$
−0.304997 + 0.952353i $$0.598656\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ −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
$$55$$ 8.00000 1.07872
$$56$$ −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 $$-0.374508\pi$$
0.384111 + 0.923287i $$0.374508\pi$$
$$62$$ 0 0
$$63$$ −1.00000 −0.125988
$$64$$ 1.00000 0.125000
$$65$$ −12.0000 −1.48842
$$66$$ 4.00000 0.492366
$$67$$ 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 $$-0.301014\pi$$
0.585206 + 0.810885i $$0.301014\pi$$
$$74$$ −10.0000 −1.16248
$$75$$ 1.00000 0.115470
$$76$$ −4.00000 −0.458831
$$77$$ 4.00000 0.455842
$$78$$ −6.00000 −0.679366
$$79$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$80$$ −2.00000 −0.223607
$$81$$ 1.00000 0.111111
$$82$$ −6.00000 −0.662589
$$83$$ −4.00000 −0.439057 −0.219529 0.975606i $$-0.570452\pi$$
−0.219529 + 0.975606i $$0.570452\pi$$
$$84$$ 1.00000 0.109109
$$85$$ −4.00000 −0.433861
$$86$$ −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 $$0.603011\pi$$
$$90$$ −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
$$99$$ −4.00000 −0.402015
$$100$$ −1.00000 −0.100000
$$101$$ −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 $$-0.371045\pi$$
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 $$-0.303036\pi$$
0.580042 + 0.814587i $$0.303036\pi$$
$$108$$ −1.00000 −0.0962250
$$109$$ −2.00000 −0.191565 −0.0957826 0.995402i $$-0.530535\pi$$
−0.0957826 + 0.995402i $$0.530535\pi$$
$$110$$ 8.00000 0.762770
$$111$$ 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
$$121$$ 5.00000 0.454545
$$122$$ 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 $$-0.838289\pi$$
−0.873704 + 0.486458i $$0.838289\pi$$
$$132$$ 4.00000 0.348155
$$133$$ 4.00000 0.346844
$$134$$ 4.00000 0.345547
$$135$$ 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 $$-0.445740\pi$$
0.169638 + 0.985506i $$0.445740\pi$$
$$140$$ 2.00000 0.169031
$$141$$ 0 0
$$142$$ 8.00000 0.671345
$$143$$ −24.0000 −2.00698
$$144$$ 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 $$-0.420959\pi$$
0.245770 + 0.969328i $$0.420959\pi$$
$$150$$ 1.00000 0.0816497
$$151$$ −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 $$-0.630658\pi$$
−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
$$165$$ −8.00000 −0.622799
$$166$$ −4.00000 −0.310460
$$167$$ −8.00000 −0.619059 −0.309529 0.950890i $$-0.600171\pi$$
−0.309529 + 0.950890i $$0.600171\pi$$
$$168$$ 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 $$-0.184706\pi$$
0.836315 + 0.548250i $$0.184706\pi$$
$$174$$ 2.00000 0.151620
$$175$$ 1.00000 0.0755929
$$176$$ −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 $$0.648028\pi$$
$$180$$ −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
$$187$$ −8.00000 −0.585018
$$188$$ 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 $$-0.477068\pi$$
0.0719816 + 0.997406i $$0.477068\pi$$
$$194$$ −14.0000 −1.00514
$$195$$ 12.0000 0.859338
$$196$$ 1.00000 0.0714286
$$197$$ −10.0000 −0.712470 −0.356235 0.934396i $$-0.615940\pi$$
−0.356235 + 0.934396i $$0.615940\pi$$
$$198$$ −4.00000 −0.284268
$$199$$ 8.00000 0.567105 0.283552 0.958957i $$-0.408487\pi$$
0.283552 + 0.958957i $$0.408487\pi$$
$$200$$ −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
$$209$$ 16.0000 1.10674
$$210$$ −2.00000 −0.138013
$$211$$ 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
$$215$$ 8.00000 0.545595
$$216$$ −1.00000 −0.0680414
$$217$$ 0 0
$$218$$ −2.00000 −0.135457
$$219$$ −10.0000 −0.675737
$$220$$ 8.00000 0.539360
$$221$$ 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 $$-0.369623\pi$$
0.398234 + 0.917284i $$0.369623\pi$$
$$228$$ 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
$$231$$ −4.00000 −0.263181
$$232$$ −2.00000 −0.131306
$$233$$ −22.0000 −1.44127 −0.720634 0.693316i $$-0.756149\pi$$
−0.720634 + 0.693316i $$0.756149\pi$$
$$234$$ 6.00000 0.392232
$$235$$ 0 0
$$236$$ 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 $$-0.479482\pi$$
0.0644157 + 0.997923i $$0.479482\pi$$
$$242$$ 5.00000 0.321412
$$243$$ −1.00000 −0.0641500
$$244$$ 6.00000 0.384111
$$245$$ −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
$$253$$ −32.0000 −2.01182
$$254$$ 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 $$-0.765152\pi$$
−0.739952 + 0.672660i $$0.765152\pi$$
$$264$$ 4.00000 0.246183
$$265$$ −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
$$275$$ 4.00000 0.241209
$$276$$ −8.00000 −0.481543
$$277$$ −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
$$280$$ 2.00000 0.119523
$$281$$ 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 $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ 8.00000 0.474713
$$285$$ −8.00000 −0.473879
$$286$$ −24.0000 −1.41915
$$287$$ 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 $$-0.159998\pi$$
0.876309 + 0.481749i $$0.159998\pi$$
$$294$$ −1.00000 −0.0583212
$$295$$ −8.00000 −0.465778
$$296$$ −10.0000 −0.581238
$$297$$ 4.00000 0.232104
$$298$$ 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 $$-0.205351\pi$$
0.799022 + 0.601302i $$0.205351\pi$$
$$308$$ 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 $$-0.408798\pi$$
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
$$319$$ 8.00000 0.447914
$$320$$ −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
$$330$$ −8.00000 −0.440386
$$331$$ −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 $$-0.336901\pi$$
0.490261 + 0.871576i $$0.336901\pi$$
$$338$$ 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 $$-0.395612\pi$$
0.322097 + 0.946707i $$0.395612\pi$$
$$348$$ 2.00000 0.107211
$$349$$ 22.0000 1.17763 0.588817 0.808267i $$-0.299594\pi$$
0.588817 + 0.808267i $$0.299594\pi$$
$$350$$ 1.00000 0.0534522
$$351$$ −6.00000 −0.320256
$$352$$ −4.00000 −0.213201
$$353$$ −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 $$-0.567708\pi$$
−0.211112 + 0.977462i $$0.567708\pi$$
$$360$$ −2.00000 −0.105409
$$361$$ −3.00000 −0.157895
$$362$$ −18.0000 −0.946059
$$363$$ −5.00000 −0.262432
$$364$$ −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 $$-0.307114\pi$$
0.569558 + 0.821951i $$0.307114\pi$$
$$374$$ −8.00000 −0.413670
$$375$$ −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
$$385$$ −8.00000 −0.407718
$$386$$ 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
$$396$$ −4.00000 −0.201008
$$397$$ 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
$$407$$ 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 $$0.683059\pi$$
$$410$$ 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
$$418$$ 16.0000 0.782586
$$419$$ −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
$$429$$ 24.0000 1.15873
$$430$$ 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 $$-0.694115\pi$$
−0.572729 + 0.819745i $$0.694115\pi$$
$$440$$ 8.00000 0.381385
$$441$$ 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
$$451$$ 24.0000 1.13012
$$452$$ −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 $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ −2.00000 −0.0934539
$$459$$ −2.00000 −0.0933520
$$460$$ −16.0000 −0.746004
$$461$$ 22.0000 1.02464 0.512321 0.858794i $$-0.328786\pi$$
0.512321 + 0.858794i $$0.328786\pi$$
$$462$$ −4.00000 −0.186097
$$463$$ −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
$$473$$ 16.0000 0.735681
$$474$$ 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 $$-0.619112\pi$$
−0.365529 + 0.930800i $$0.619112\pi$$
$$480$$ 2.00000 0.0912871
$$481$$ −60.0000 −2.73576
$$482$$ 2.00000 0.0910975
$$483$$ 8.00000 0.364013
$$484$$ 5.00000 0.227273
$$485$$ 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 $$-0.412720\pi$$
0.270776 + 0.962642i $$0.412720\pi$$
$$492$$ 6.00000 0.270501
$$493$$ −4.00000 −0.180151
$$494$$ −24.0000 −1.07981
$$495$$ 8.00000 0.359573
$$496$$ 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 $$-0.320291\pi$$
0.535054 + 0.844818i $$0.320291\pi$$
$$504$$ −1.00000 −0.0445435
$$505$$ 4.00000 0.177998
$$506$$ −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 $$0.457548\pi$$
$$510$$ 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 $$-0.355946\pi$$
0.437269 + 0.899331i $$0.355946\pi$$
$$524$$ −20.0000 −0.873704
$$525$$ −1.00000 −0.0436436
$$526$$ −24.0000 −1.04645
$$527$$ 0 0
$$528$$ 4.00000 0.174078
$$529$$ 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
$$539$$ −4.00000 −0.172292
$$540$$ 2.00000 0.0860663
$$541$$ 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
$$545$$ 4.00000 0.171341
$$546$$ 6.00000 0.256776
$$547$$ −12.0000 −0.513083 −0.256541 0.966533i $$-0.582583\pi$$
−0.256541 + 0.966533i $$0.582583\pi$$
$$548$$ 10.0000 0.427179
$$549$$ 6.00000 0.256074
$$550$$ 4.00000 0.170561
$$551$$ 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 $$-0.513491\pi$$
−0.0423714 + 0.999102i $$0.513491\pi$$
$$558$$ 0 0
$$559$$ −24.0000 −1.01509
$$560$$ 2.00000 0.0845154
$$561$$ 8.00000 0.337760
$$562$$ 26.0000 1.09674
$$563$$ 44.0000 1.85438 0.927189 0.374593i $$-0.122217\pi$$
0.927189 + 0.374593i $$0.122217\pi$$
$$564$$ 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 $$-0.540139\pi$$
−0.125767 + 0.992060i $$0.540139\pi$$
$$570$$ −8.00000 −0.335083
$$571$$ 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
$$575$$ −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
$$583$$ −24.0000 −0.993978
$$584$$ 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 $$-0.379500\pi$$
0.369586 + 0.929197i $$0.379500\pi$$
$$594$$ 4.00000 0.164122
$$595$$ 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 $$-0.322086\pi$$
0.530281 + 0.847822i $$0.322086\pi$$
$$602$$ 4.00000 0.163028
$$603$$ 4.00000 0.162893
$$604$$ −8.00000 −0.325515
$$605$$ −10.0000 −0.406558
$$606$$ 2.00000 0.0812444
$$607$$ 48.0000 1.94826 0.974130 0.225989i $$-0.0725612\pi$$
0.974130 + 0.225989i $$0.0725612\pi$$
$$608$$ −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 $$-0.822303\pi$$
−0.848182 + 0.529705i $$0.822303\pi$$
$$614$$ 28.0000 1.12999
$$615$$ −12.0000 −0.483887
$$616$$ 4.00000 0.161165
$$617$$ −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
$$627$$ −16.0000 −0.638978
$$628$$ −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
$$638$$ 8.00000 0.316723
$$639$$ 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
$$649$$ −16.0000 −0.628055
$$650$$ −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 $$-0.683608\pi$$
−0.545363 + 0.838200i $$0.683608\pi$$
$$660$$ −8.00000 −0.311400
$$661$$ −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
$$671$$ −24.0000 −0.926510
$$672$$ 1.00000 0.0385758
$$673$$ 2.00000 0.0770943 0.0385472 0.999257i $$-0.487727\pi$$
0.0385472 + 0.999257i $$0.487727\pi$$
$$674$$ 18.0000 0.693334
$$675$$ 1.00000 0.0384900
$$676$$ 23.0000 0.884615
$$677$$ −18.0000 −0.691796 −0.345898 0.938272i $$-0.612426\pi$$
−0.345898 + 0.938272i $$0.612426\pi$$
$$678$$ 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
$$693$$ 4.00000 0.151947
$$694$$ 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 $$-0.512025\pi$$
−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 $$0.560129\pi$$
$$710$$ −16.0000 −0.600469
$$711$$ 0 0
$$712$$ −6.00000 −0.224860
$$713$$ 0 0
$$714$$ 2.00000 0.0748481
$$715$$ 48.0000 1.79510
$$716$$ −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
$$726$$ −5.00000 −0.185567
$$727$$ −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 $$-0.464656\pi$$
0.110808 + 0.993842i $$0.464656\pi$$
$$734$$ 32.0000 1.18114
$$735$$ 2.00000 0.0737711
$$736$$ 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 $$0.570839\pi$$
$$740$$ 20.0000 0.735215
$$741$$ 24.0000 0.881662
$$742$$ −6.00000 −0.220267
$$743$$ 24.0000 0.880475 0.440237 0.897881i $$-0.354894\pi$$
0.440237 + 0.897881i $$0.354894\pi$$
$$744$$ 0 0
$$745$$ −12.0000 −0.439646
$$746$$ 22.0000 0.805477
$$747$$ −4.00000 −0.146352
$$748$$ −8.00000 −0.292509
$$749$$ −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
$$759$$ 32.0000 1.16153
$$760$$ 8.00000 0.290191
$$761$$ −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
$$765$$ −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 $$-0.581229\pi$$
−0.252426 + 0.967616i $$0.581229\pi$$
$$770$$ −8.00000 −0.288300
$$771$$ 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
$$781$$ −32.0000 −1.14505
$$782$$ 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 $$-0.721742\pi$$
−0.641631 + 0.767014i $$0.721742\pi$$
$$788$$ −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
$$795$$ 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
$$803$$ −40.0000 −1.41157
$$804$$ −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 $$-0.443752\pi$$
0.175791 + 0.984428i $$0.443752\pi$$
$$810$$ −2.00000 −0.0702728
$$811$$ −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
$$815$$ −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 $$-0.269238\pi$$
0.663105 + 0.748527i $$0.269238\pi$$
$$822$$ −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
$$825$$ −4.00000 −0.139262
$$826$$ −4.00000 −0.139178
$$827$$ −36.0000 −1.25184 −0.625921 0.779886i $$-0.715277\pi$$
−0.625921 + 0.779886i $$0.715277\pi$$
$$828$$ 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
$$836$$ 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 $$0.0824164\pi$$
$$840$$ −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
$$847$$ −5.00000 −0.171802
$$848$$ 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 $$-0.422959\pi$$
0.239675 + 0.970853i $$0.422959\pi$$
$$854$$ −6.00000 −0.205316
$$855$$ 8.00000 0.273594
$$856$$ 12.0000 0.410152
$$857$$ 42.0000 1.43469 0.717346 0.696717i $$-0.245357\pi$$
0.717346 + 0.696717i $$0.245357\pi$$
$$858$$ 24.0000 0.819346
$$859$$ 20.0000 0.682391 0.341196 0.939992i $$-0.389168\pi$$
0.341196 + 0.939992i $$0.389168\pi$$
$$860$$ 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 $$-0.510751\pi$$
−0.0337676 + 0.999430i $$0.510751\pi$$
$$878$$ −24.0000 −0.809961
$$879$$ −30.0000 −1.01187
$$880$$ 8.00000 0.269680
$$881$$ 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 $$-0.632005\pi$$
−0.402921 + 0.915235i $$0.632005\pi$$
$$888$$ 10.0000 0.335578
$$889$$ 0 0
$$890$$ 12.0000 0.402241
$$891$$ −4.00000 −0.134005
$$892$$ −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
$$902$$ 24.0000 0.799113
$$903$$ −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
$$913$$ 16.0000 0.529523
$$914$$ 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 $$-0.270667\pi$$
0.659739 + 0.751495i $$0.270667\pi$$
$$920$$ −16.0000 −0.527504
$$921$$ −28.0000 −0.922631
$$922$$ 22.0000 0.724531
$$923$$ 48.0000 1.57994
$$924$$ −4.00000 −0.131590
$$925$$ 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
$$935$$ 16.0000 0.523256
$$936$$ 6.00000 0.196116
$$937$$ −22.0000 −0.718709 −0.359354 0.933201i $$-0.617003\pi$$
−0.359354 + 0.933201i $$0.617003\pi$$
$$938$$ −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
$$945$$ −2.00000 −0.0650600
$$946$$ 16.0000 0.520205
$$947$$ 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 $$-0.361640\pi$$
0.421111 + 0.907009i $$0.361640\pi$$
$$954$$ 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
$$960$$ 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 $$-0.458942\pi$$
0.128631 + 0.991692i $$0.458942\pi$$
$$968$$ 5.00000 0.160706
$$969$$ 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
$$979$$ 24.0000 0.767043
$$980$$ −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
$$990$$ 8.00000 0.254257
$$991$$ −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 $$-0.428842\pi$$
0.221692 + 0.975117i $$0.428842\pi$$
$$998$$ −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 42.2.a.a.1.1 1
3.2 odd 2 126.2.a.a.1.1 1
4.3 odd 2 336.2.a.d.1.1 1
5.2 odd 4 1050.2.g.a.799.2 2
5.3 odd 4 1050.2.g.a.799.1 2
5.4 even 2 1050.2.a.i.1.1 1
7.2 even 3 294.2.e.c.67.1 2
7.3 odd 6 294.2.e.a.79.1 2
7.4 even 3 294.2.e.c.79.1 2
7.5 odd 6 294.2.e.a.67.1 2
7.6 odd 2 294.2.a.g.1.1 1
8.3 odd 2 1344.2.a.i.1.1 1
8.5 even 2 1344.2.a.q.1.1 1
9.2 odd 6 1134.2.f.j.757.1 2
9.4 even 3 1134.2.f.g.379.1 2
9.5 odd 6 1134.2.f.j.379.1 2
9.7 even 3 1134.2.f.g.757.1 2
11.10 odd 2 5082.2.a.d.1.1 1
12.11 even 2 1008.2.a.j.1.1 1
13.12 even 2 7098.2.a.f.1.1 1
15.2 even 4 3150.2.g.r.2899.1 2
15.8 even 4 3150.2.g.r.2899.2 2
15.14 odd 2 3150.2.a.bo.1.1 1
16.3 odd 4 5376.2.c.e.2689.2 2
16.5 even 4 5376.2.c.bc.2689.2 2
16.11 odd 4 5376.2.c.e.2689.1 2
16.13 even 4 5376.2.c.bc.2689.1 2
20.19 odd 2 8400.2.a.k.1.1 1
21.2 odd 6 882.2.g.h.361.1 2
21.5 even 6 882.2.g.j.361.1 2
21.11 odd 6 882.2.g.h.667.1 2
21.17 even 6 882.2.g.j.667.1 2
21.20 even 2 882.2.a.b.1.1 1
24.5 odd 2 4032.2.a.e.1.1 1
24.11 even 2 4032.2.a.m.1.1 1
28.3 even 6 2352.2.q.n.961.1 2
28.11 odd 6 2352.2.q.i.961.1 2
28.19 even 6 2352.2.q.n.1537.1 2
28.23 odd 6 2352.2.q.i.1537.1 2
28.27 even 2 2352.2.a.l.1.1 1
35.34 odd 2 7350.2.a.f.1.1 1
56.13 odd 2 9408.2.a.n.1.1 1
56.27 even 2 9408.2.a.bw.1.1 1
84.83 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 1.1 even 1 trivial
126.2.a.a.1.1 1 3.2 odd 2
294.2.a.g.1.1 1 7.6 odd 2
294.2.e.a.67.1 2 7.5 odd 6
294.2.e.a.79.1 2 7.3 odd 6
294.2.e.c.67.1 2 7.2 even 3
294.2.e.c.79.1 2 7.4 even 3
336.2.a.d.1.1 1 4.3 odd 2
882.2.a.b.1.1 1 21.20 even 2
882.2.g.h.361.1 2 21.2 odd 6
882.2.g.h.667.1 2 21.11 odd 6
882.2.g.j.361.1 2 21.5 even 6
882.2.g.j.667.1 2 21.17 even 6
1008.2.a.j.1.1 1 12.11 even 2
1050.2.a.i.1.1 1 5.4 even 2
1050.2.g.a.799.1 2 5.3 odd 4
1050.2.g.a.799.2 2 5.2 odd 4
1134.2.f.g.379.1 2 9.4 even 3
1134.2.f.g.757.1 2 9.7 even 3
1134.2.f.j.379.1 2 9.5 odd 6
1134.2.f.j.757.1 2 9.2 odd 6
1344.2.a.i.1.1 1 8.3 odd 2
1344.2.a.q.1.1 1 8.5 even 2
2352.2.a.l.1.1 1 28.27 even 2
2352.2.q.i.961.1 2 28.11 odd 6
2352.2.q.i.1537.1 2 28.23 odd 6
2352.2.q.n.961.1 2 28.3 even 6
2352.2.q.n.1537.1 2 28.19 even 6
3150.2.a.bo.1.1 1 15.14 odd 2
3150.2.g.r.2899.1 2 15.2 even 4
3150.2.g.r.2899.2 2 15.8 even 4
4032.2.a.e.1.1 1 24.5 odd 2
4032.2.a.m.1.1 1 24.11 even 2
5082.2.a.d.1.1 1 11.10 odd 2
5376.2.c.e.2689.1 2 16.11 odd 4
5376.2.c.e.2689.2 2 16.3 odd 4
5376.2.c.bc.2689.1 2 16.13 even 4
5376.2.c.bc.2689.2 2 16.5 even 4
7056.2.a.k.1.1 1 84.83 odd 2
7098.2.a.f.1.1 1 13.12 even 2
7350.2.a.f.1.1 1 35.34 odd 2
8400.2.a.k.1.1 1 20.19 odd 2
9408.2.a.n.1.1 1 56.13 odd 2
9408.2.a.bw.1.1 1 56.27 even 2