# Properties

 Label 690.2.a.e.1.1 Level $690$ Weight $2$ Character 690.1 Self dual yes Analytic conductor $5.510$ Analytic rank $1$ Dimension $1$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$690 = 2 \cdot 3 \cdot 5 \cdot 23$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 690.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$5.50967773947$$ Analytic rank: $$1$$ 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 $$\chi$$ $$=$$ 690.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} -1.00000 q^{6} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} -4.00000 q^{11} +1.00000 q^{12} -6.00000 q^{13} -1.00000 q^{15} +1.00000 q^{16} -6.00000 q^{17} -1.00000 q^{18} +4.00000 q^{19} -1.00000 q^{20} +4.00000 q^{22} +1.00000 q^{23} -1.00000 q^{24} +1.00000 q^{25} +6.00000 q^{26} +1.00000 q^{27} -6.00000 q^{29} +1.00000 q^{30} -8.00000 q^{31} -1.00000 q^{32} -4.00000 q^{33} +6.00000 q^{34} +1.00000 q^{36} +6.00000 q^{37} -4.00000 q^{38} -6.00000 q^{39} +1.00000 q^{40} +10.0000 q^{41} +4.00000 q^{43} -4.00000 q^{44} -1.00000 q^{45} -1.00000 q^{46} -8.00000 q^{47} +1.00000 q^{48} -7.00000 q^{49} -1.00000 q^{50} -6.00000 q^{51} -6.00000 q^{52} -14.0000 q^{53} -1.00000 q^{54} +4.00000 q^{55} +4.00000 q^{57} +6.00000 q^{58} -1.00000 q^{60} +10.0000 q^{61} +8.00000 q^{62} +1.00000 q^{64} +6.00000 q^{65} +4.00000 q^{66} +4.00000 q^{67} -6.00000 q^{68} +1.00000 q^{69} +8.00000 q^{71} -1.00000 q^{72} +2.00000 q^{73} -6.00000 q^{74} +1.00000 q^{75} +4.00000 q^{76} +6.00000 q^{78} -12.0000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -10.0000 q^{82} -16.0000 q^{83} +6.00000 q^{85} -4.00000 q^{86} -6.00000 q^{87} +4.00000 q^{88} -2.00000 q^{89} +1.00000 q^{90} +1.00000 q^{92} -8.00000 q^{93} +8.00000 q^{94} -4.00000 q^{95} -1.00000 q^{96} -14.0000 q^{97} +7.00000 q^{98} -4.00000 q^{99} +O(q^{100})$$

## 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
$$5$$ −1.00000 −0.447214
$$6$$ −1.00000 −0.408248
$$7$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$8$$ −1.00000 −0.353553
$$9$$ 1.00000 0.333333
$$10$$ 1.00000 0.316228
$$11$$ −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 $$0.812833\pi$$
$$14$$ 0 0
$$15$$ −1.00000 −0.258199
$$16$$ 1.00000 0.250000
$$17$$ −6.00000 −1.45521 −0.727607 0.685994i $$-0.759367\pi$$
−0.727607 + 0.685994i $$0.759367\pi$$
$$18$$ −1.00000 −0.235702
$$19$$ 4.00000 0.917663 0.458831 0.888523i $$-0.348268\pi$$
0.458831 + 0.888523i $$0.348268\pi$$
$$20$$ −1.00000 −0.223607
$$21$$ 0 0
$$22$$ 4.00000 0.852803
$$23$$ 1.00000 0.208514
$$24$$ −1.00000 −0.204124
$$25$$ 1.00000 0.200000
$$26$$ 6.00000 1.17670
$$27$$ 1.00000 0.192450
$$28$$ 0 0
$$29$$ −6.00000 −1.11417 −0.557086 0.830455i $$-0.688081\pi$$
−0.557086 + 0.830455i $$0.688081\pi$$
$$30$$ 1.00000 0.182574
$$31$$ −8.00000 −1.43684 −0.718421 0.695608i $$-0.755135\pi$$
−0.718421 + 0.695608i $$0.755135\pi$$
$$32$$ −1.00000 −0.176777
$$33$$ −4.00000 −0.696311
$$34$$ 6.00000 1.02899
$$35$$ 0 0
$$36$$ 1.00000 0.166667
$$37$$ 6.00000 0.986394 0.493197 0.869918i $$-0.335828\pi$$
0.493197 + 0.869918i $$0.335828\pi$$
$$38$$ −4.00000 −0.648886
$$39$$ −6.00000 −0.960769
$$40$$ 1.00000 0.158114
$$41$$ 10.0000 1.56174 0.780869 0.624695i $$-0.214777\pi$$
0.780869 + 0.624695i $$0.214777\pi$$
$$42$$ 0 0
$$43$$ 4.00000 0.609994 0.304997 0.952353i $$-0.401344\pi$$
0.304997 + 0.952353i $$0.401344\pi$$
$$44$$ −4.00000 −0.603023
$$45$$ −1.00000 −0.149071
$$46$$ −1.00000 −0.147442
$$47$$ −8.00000 −1.16692 −0.583460 0.812142i $$-0.698301\pi$$
−0.583460 + 0.812142i $$0.698301\pi$$
$$48$$ 1.00000 0.144338
$$49$$ −7.00000 −1.00000
$$50$$ −1.00000 −0.141421
$$51$$ −6.00000 −0.840168
$$52$$ −6.00000 −0.832050
$$53$$ −14.0000 −1.92305 −0.961524 0.274721i $$-0.911414\pi$$
−0.961524 + 0.274721i $$0.911414\pi$$
$$54$$ −1.00000 −0.136083
$$55$$ 4.00000 0.539360
$$56$$ 0 0
$$57$$ 4.00000 0.529813
$$58$$ 6.00000 0.787839
$$59$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$60$$ −1.00000 −0.129099
$$61$$ 10.0000 1.28037 0.640184 0.768221i $$-0.278858\pi$$
0.640184 + 0.768221i $$0.278858\pi$$
$$62$$ 8.00000 1.01600
$$63$$ 0 0
$$64$$ 1.00000 0.125000
$$65$$ 6.00000 0.744208
$$66$$ 4.00000 0.492366
$$67$$ 4.00000 0.488678 0.244339 0.969690i $$-0.421429\pi$$
0.244339 + 0.969690i $$0.421429\pi$$
$$68$$ −6.00000 −0.727607
$$69$$ 1.00000 0.120386
$$70$$ 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
$$73$$ 2.00000 0.234082 0.117041 0.993127i $$-0.462659\pi$$
0.117041 + 0.993127i $$0.462659\pi$$
$$74$$ −6.00000 −0.697486
$$75$$ 1.00000 0.115470
$$76$$ 4.00000 0.458831
$$77$$ 0 0
$$78$$ 6.00000 0.679366
$$79$$ −12.0000 −1.35011 −0.675053 0.737769i $$-0.735879\pi$$
−0.675053 + 0.737769i $$0.735879\pi$$
$$80$$ −1.00000 −0.111803
$$81$$ 1.00000 0.111111
$$82$$ −10.0000 −1.10432
$$83$$ −16.0000 −1.75623 −0.878114 0.478451i $$-0.841198\pi$$
−0.878114 + 0.478451i $$0.841198\pi$$
$$84$$ 0 0
$$85$$ 6.00000 0.650791
$$86$$ −4.00000 −0.431331
$$87$$ −6.00000 −0.643268
$$88$$ 4.00000 0.426401
$$89$$ −2.00000 −0.212000 −0.106000 0.994366i $$-0.533804\pi$$
−0.106000 + 0.994366i $$0.533804\pi$$
$$90$$ 1.00000 0.105409
$$91$$ 0 0
$$92$$ 1.00000 0.104257
$$93$$ −8.00000 −0.829561
$$94$$ 8.00000 0.825137
$$95$$ −4.00000 −0.410391
$$96$$ −1.00000 −0.102062
$$97$$ −14.0000 −1.42148 −0.710742 0.703452i $$-0.751641\pi$$
−0.710742 + 0.703452i $$0.751641\pi$$
$$98$$ 7.00000 0.707107
$$99$$ −4.00000 −0.402015
$$100$$ 1.00000 0.100000
$$101$$ 18.0000 1.79107 0.895533 0.444994i $$-0.146794\pi$$
0.895533 + 0.444994i $$0.146794\pi$$
$$102$$ 6.00000 0.594089
$$103$$ −16.0000 −1.57653 −0.788263 0.615338i $$-0.789020\pi$$
−0.788263 + 0.615338i $$0.789020\pi$$
$$104$$ 6.00000 0.588348
$$105$$ 0 0
$$106$$ 14.0000 1.35980
$$107$$ 8.00000 0.773389 0.386695 0.922208i $$-0.373617\pi$$
0.386695 + 0.922208i $$0.373617\pi$$
$$108$$ 1.00000 0.0962250
$$109$$ 2.00000 0.191565 0.0957826 0.995402i $$-0.469465\pi$$
0.0957826 + 0.995402i $$0.469465\pi$$
$$110$$ −4.00000 −0.381385
$$111$$ 6.00000 0.569495
$$112$$ 0 0
$$113$$ −14.0000 −1.31701 −0.658505 0.752577i $$-0.728811\pi$$
−0.658505 + 0.752577i $$0.728811\pi$$
$$114$$ −4.00000 −0.374634
$$115$$ −1.00000 −0.0932505
$$116$$ −6.00000 −0.557086
$$117$$ −6.00000 −0.554700
$$118$$ 0 0
$$119$$ 0 0
$$120$$ 1.00000 0.0912871
$$121$$ 5.00000 0.454545
$$122$$ −10.0000 −0.905357
$$123$$ 10.0000 0.901670
$$124$$ −8.00000 −0.718421
$$125$$ −1.00000 −0.0894427
$$126$$ 0 0
$$127$$ 12.0000 1.06483 0.532414 0.846484i $$-0.321285\pi$$
0.532414 + 0.846484i $$0.321285\pi$$
$$128$$ −1.00000 −0.0883883
$$129$$ 4.00000 0.352180
$$130$$ −6.00000 −0.526235
$$131$$ 8.00000 0.698963 0.349482 0.936943i $$-0.386358\pi$$
0.349482 + 0.936943i $$0.386358\pi$$
$$132$$ −4.00000 −0.348155
$$133$$ 0 0
$$134$$ −4.00000 −0.345547
$$135$$ −1.00000 −0.0860663
$$136$$ 6.00000 0.514496
$$137$$ 2.00000 0.170872 0.0854358 0.996344i $$-0.472772\pi$$
0.0854358 + 0.996344i $$0.472772\pi$$
$$138$$ −1.00000 −0.0851257
$$139$$ 12.0000 1.01783 0.508913 0.860818i $$-0.330047\pi$$
0.508913 + 0.860818i $$0.330047\pi$$
$$140$$ 0 0
$$141$$ −8.00000 −0.673722
$$142$$ −8.00000 −0.671345
$$143$$ 24.0000 2.00698
$$144$$ 1.00000 0.0833333
$$145$$ 6.00000 0.498273
$$146$$ −2.00000 −0.165521
$$147$$ −7.00000 −0.577350
$$148$$ 6.00000 0.493197
$$149$$ 10.0000 0.819232 0.409616 0.912258i $$-0.365663\pi$$
0.409616 + 0.912258i $$0.365663\pi$$
$$150$$ −1.00000 −0.0816497
$$151$$ 16.0000 1.30206 0.651031 0.759051i $$-0.274337\pi$$
0.651031 + 0.759051i $$0.274337\pi$$
$$152$$ −4.00000 −0.324443
$$153$$ −6.00000 −0.485071
$$154$$ 0 0
$$155$$ 8.00000 0.642575
$$156$$ −6.00000 −0.480384
$$157$$ 6.00000 0.478852 0.239426 0.970915i $$-0.423041\pi$$
0.239426 + 0.970915i $$0.423041\pi$$
$$158$$ 12.0000 0.954669
$$159$$ −14.0000 −1.11027
$$160$$ 1.00000 0.0790569
$$161$$ 0 0
$$162$$ −1.00000 −0.0785674
$$163$$ 4.00000 0.313304 0.156652 0.987654i $$-0.449930\pi$$
0.156652 + 0.987654i $$0.449930\pi$$
$$164$$ 10.0000 0.780869
$$165$$ 4.00000 0.311400
$$166$$ 16.0000 1.24184
$$167$$ −16.0000 −1.23812 −0.619059 0.785345i $$-0.712486\pi$$
−0.619059 + 0.785345i $$0.712486\pi$$
$$168$$ 0 0
$$169$$ 23.0000 1.76923
$$170$$ −6.00000 −0.460179
$$171$$ 4.00000 0.305888
$$172$$ 4.00000 0.304997
$$173$$ 18.0000 1.36851 0.684257 0.729241i $$-0.260127\pi$$
0.684257 + 0.729241i $$0.260127\pi$$
$$174$$ 6.00000 0.454859
$$175$$ 0 0
$$176$$ −4.00000 −0.301511
$$177$$ 0 0
$$178$$ 2.00000 0.149906
$$179$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$180$$ −1.00000 −0.0745356
$$181$$ −6.00000 −0.445976 −0.222988 0.974821i $$-0.571581\pi$$
−0.222988 + 0.974821i $$0.571581\pi$$
$$182$$ 0 0
$$183$$ 10.0000 0.739221
$$184$$ −1.00000 −0.0737210
$$185$$ −6.00000 −0.441129
$$186$$ 8.00000 0.586588
$$187$$ 24.0000 1.75505
$$188$$ −8.00000 −0.583460
$$189$$ 0 0
$$190$$ 4.00000 0.290191
$$191$$ −8.00000 −0.578860 −0.289430 0.957199i $$-0.593466\pi$$
−0.289430 + 0.957199i $$0.593466\pi$$
$$192$$ 1.00000 0.0721688
$$193$$ 10.0000 0.719816 0.359908 0.932988i $$-0.382808\pi$$
0.359908 + 0.932988i $$0.382808\pi$$
$$194$$ 14.0000 1.00514
$$195$$ 6.00000 0.429669
$$196$$ −7.00000 −0.500000
$$197$$ 26.0000 1.85242 0.926212 0.377004i $$-0.123046\pi$$
0.926212 + 0.377004i $$0.123046\pi$$
$$198$$ 4.00000 0.284268
$$199$$ −12.0000 −0.850657 −0.425329 0.905039i $$-0.639842\pi$$
−0.425329 + 0.905039i $$0.639842\pi$$
$$200$$ −1.00000 −0.0707107
$$201$$ 4.00000 0.282138
$$202$$ −18.0000 −1.26648
$$203$$ 0 0
$$204$$ −6.00000 −0.420084
$$205$$ −10.0000 −0.698430
$$206$$ 16.0000 1.11477
$$207$$ 1.00000 0.0695048
$$208$$ −6.00000 −0.416025
$$209$$ −16.0000 −1.10674
$$210$$ 0 0
$$211$$ −20.0000 −1.37686 −0.688428 0.725304i $$-0.741699\pi$$
−0.688428 + 0.725304i $$0.741699\pi$$
$$212$$ −14.0000 −0.961524
$$213$$ 8.00000 0.548151
$$214$$ −8.00000 −0.546869
$$215$$ −4.00000 −0.272798
$$216$$ −1.00000 −0.0680414
$$217$$ 0 0
$$218$$ −2.00000 −0.135457
$$219$$ 2.00000 0.135147
$$220$$ 4.00000 0.269680
$$221$$ 36.0000 2.42162
$$222$$ −6.00000 −0.402694
$$223$$ −20.0000 −1.33930 −0.669650 0.742677i $$-0.733556\pi$$
−0.669650 + 0.742677i $$0.733556\pi$$
$$224$$ 0 0
$$225$$ 1.00000 0.0666667
$$226$$ 14.0000 0.931266
$$227$$ 8.00000 0.530979 0.265489 0.964114i $$-0.414466\pi$$
0.265489 + 0.964114i $$0.414466\pi$$
$$228$$ 4.00000 0.264906
$$229$$ −14.0000 −0.925146 −0.462573 0.886581i $$-0.653074\pi$$
−0.462573 + 0.886581i $$0.653074\pi$$
$$230$$ 1.00000 0.0659380
$$231$$ 0 0
$$232$$ 6.00000 0.393919
$$233$$ 22.0000 1.44127 0.720634 0.693316i $$-0.243851\pi$$
0.720634 + 0.693316i $$0.243851\pi$$
$$234$$ 6.00000 0.392232
$$235$$ 8.00000 0.521862
$$236$$ 0 0
$$237$$ −12.0000 −0.779484
$$238$$ 0 0
$$239$$ −8.00000 −0.517477 −0.258738 0.965947i $$-0.583307\pi$$
−0.258738 + 0.965947i $$0.583307\pi$$
$$240$$ −1.00000 −0.0645497
$$241$$ 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
$$244$$ 10.0000 0.640184
$$245$$ 7.00000 0.447214
$$246$$ −10.0000 −0.637577
$$247$$ −24.0000 −1.52708
$$248$$ 8.00000 0.508001
$$249$$ −16.0000 −1.01396
$$250$$ 1.00000 0.0632456
$$251$$ −12.0000 −0.757433 −0.378717 0.925513i $$-0.623635\pi$$
−0.378717 + 0.925513i $$0.623635\pi$$
$$252$$ 0 0
$$253$$ −4.00000 −0.251478
$$254$$ −12.0000 −0.752947
$$255$$ 6.00000 0.375735
$$256$$ 1.00000 0.0625000
$$257$$ 6.00000 0.374270 0.187135 0.982334i $$-0.440080\pi$$
0.187135 + 0.982334i $$0.440080\pi$$
$$258$$ −4.00000 −0.249029
$$259$$ 0 0
$$260$$ 6.00000 0.372104
$$261$$ −6.00000 −0.371391
$$262$$ −8.00000 −0.494242
$$263$$ −32.0000 −1.97320 −0.986602 0.163144i $$-0.947836\pi$$
−0.986602 + 0.163144i $$0.947836\pi$$
$$264$$ 4.00000 0.246183
$$265$$ 14.0000 0.860013
$$266$$ 0 0
$$267$$ −2.00000 −0.122398
$$268$$ 4.00000 0.244339
$$269$$ 10.0000 0.609711 0.304855 0.952399i $$-0.401392\pi$$
0.304855 + 0.952399i $$0.401392\pi$$
$$270$$ 1.00000 0.0608581
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ −6.00000 −0.363803
$$273$$ 0 0
$$274$$ −2.00000 −0.120824
$$275$$ −4.00000 −0.241209
$$276$$ 1.00000 0.0601929
$$277$$ −30.0000 −1.80253 −0.901263 0.433273i $$-0.857359\pi$$
−0.901263 + 0.433273i $$0.857359\pi$$
$$278$$ −12.0000 −0.719712
$$279$$ −8.00000 −0.478947
$$280$$ 0 0
$$281$$ 30.0000 1.78965 0.894825 0.446417i $$-0.147300\pi$$
0.894825 + 0.446417i $$0.147300\pi$$
$$282$$ 8.00000 0.476393
$$283$$ 4.00000 0.237775 0.118888 0.992908i $$-0.462067\pi$$
0.118888 + 0.992908i $$0.462067\pi$$
$$284$$ 8.00000 0.474713
$$285$$ −4.00000 −0.236940
$$286$$ −24.0000 −1.41915
$$287$$ 0 0
$$288$$ −1.00000 −0.0589256
$$289$$ 19.0000 1.11765
$$290$$ −6.00000 −0.352332
$$291$$ −14.0000 −0.820695
$$292$$ 2.00000 0.117041
$$293$$ −14.0000 −0.817889 −0.408944 0.912559i $$-0.634103\pi$$
−0.408944 + 0.912559i $$0.634103\pi$$
$$294$$ 7.00000 0.408248
$$295$$ 0 0
$$296$$ −6.00000 −0.348743
$$297$$ −4.00000 −0.232104
$$298$$ −10.0000 −0.579284
$$299$$ −6.00000 −0.346989
$$300$$ 1.00000 0.0577350
$$301$$ 0 0
$$302$$ −16.0000 −0.920697
$$303$$ 18.0000 1.03407
$$304$$ 4.00000 0.229416
$$305$$ −10.0000 −0.572598
$$306$$ 6.00000 0.342997
$$307$$ 12.0000 0.684876 0.342438 0.939540i $$-0.388747\pi$$
0.342438 + 0.939540i $$0.388747\pi$$
$$308$$ 0 0
$$309$$ −16.0000 −0.910208
$$310$$ −8.00000 −0.454369
$$311$$ −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 $$0.408798\pi$$
$$314$$ −6.00000 −0.338600
$$315$$ 0 0
$$316$$ −12.0000 −0.675053
$$317$$ 18.0000 1.01098 0.505490 0.862832i $$-0.331312\pi$$
0.505490 + 0.862832i $$0.331312\pi$$
$$318$$ 14.0000 0.785081
$$319$$ 24.0000 1.34374
$$320$$ −1.00000 −0.0559017
$$321$$ 8.00000 0.446516
$$322$$ 0 0
$$323$$ −24.0000 −1.33540
$$324$$ 1.00000 0.0555556
$$325$$ −6.00000 −0.332820
$$326$$ −4.00000 −0.221540
$$327$$ 2.00000 0.110600
$$328$$ −10.0000 −0.552158
$$329$$ 0 0
$$330$$ −4.00000 −0.220193
$$331$$ −20.0000 −1.09930 −0.549650 0.835395i $$-0.685239\pi$$
−0.549650 + 0.835395i $$0.685239\pi$$
$$332$$ −16.0000 −0.878114
$$333$$ 6.00000 0.328798
$$334$$ 16.0000 0.875481
$$335$$ −4.00000 −0.218543
$$336$$ 0 0
$$337$$ −14.0000 −0.762629 −0.381314 0.924445i $$-0.624528\pi$$
−0.381314 + 0.924445i $$0.624528\pi$$
$$338$$ −23.0000 −1.25104
$$339$$ −14.0000 −0.760376
$$340$$ 6.00000 0.325396
$$341$$ 32.0000 1.73290
$$342$$ −4.00000 −0.216295
$$343$$ 0 0
$$344$$ −4.00000 −0.215666
$$345$$ −1.00000 −0.0538382
$$346$$ −18.0000 −0.967686
$$347$$ −28.0000 −1.50312 −0.751559 0.659665i $$-0.770698\pi$$
−0.751559 + 0.659665i $$0.770698\pi$$
$$348$$ −6.00000 −0.321634
$$349$$ −34.0000 −1.81998 −0.909989 0.414632i $$-0.863910\pi$$
−0.909989 + 0.414632i $$0.863910\pi$$
$$350$$ 0 0
$$351$$ −6.00000 −0.320256
$$352$$ 4.00000 0.213201
$$353$$ 6.00000 0.319348 0.159674 0.987170i $$-0.448956\pi$$
0.159674 + 0.987170i $$0.448956\pi$$
$$354$$ 0 0
$$355$$ −8.00000 −0.424596
$$356$$ −2.00000 −0.106000
$$357$$ 0 0
$$358$$ 0 0
$$359$$ −8.00000 −0.422224 −0.211112 0.977462i $$-0.567708\pi$$
−0.211112 + 0.977462i $$0.567708\pi$$
$$360$$ 1.00000 0.0527046
$$361$$ −3.00000 −0.157895
$$362$$ 6.00000 0.315353
$$363$$ 5.00000 0.262432
$$364$$ 0 0
$$365$$ −2.00000 −0.104685
$$366$$ −10.0000 −0.522708
$$367$$ −32.0000 −1.67039 −0.835193 0.549957i $$-0.814644\pi$$
−0.835193 + 0.549957i $$0.814644\pi$$
$$368$$ 1.00000 0.0521286
$$369$$ 10.0000 0.520579
$$370$$ 6.00000 0.311925
$$371$$ 0 0
$$372$$ −8.00000 −0.414781
$$373$$ −18.0000 −0.932005 −0.466002 0.884783i $$-0.654306\pi$$
−0.466002 + 0.884783i $$0.654306\pi$$
$$374$$ −24.0000 −1.24101
$$375$$ −1.00000 −0.0516398
$$376$$ 8.00000 0.412568
$$377$$ 36.0000 1.85409
$$378$$ 0 0
$$379$$ 4.00000 0.205466 0.102733 0.994709i $$-0.467241\pi$$
0.102733 + 0.994709i $$0.467241\pi$$
$$380$$ −4.00000 −0.205196
$$381$$ 12.0000 0.614779
$$382$$ 8.00000 0.409316
$$383$$ 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
$$386$$ −10.0000 −0.508987
$$387$$ 4.00000 0.203331
$$388$$ −14.0000 −0.710742
$$389$$ −6.00000 −0.304212 −0.152106 0.988364i $$-0.548606\pi$$
−0.152106 + 0.988364i $$0.548606\pi$$
$$390$$ −6.00000 −0.303822
$$391$$ −6.00000 −0.303433
$$392$$ 7.00000 0.353553
$$393$$ 8.00000 0.403547
$$394$$ −26.0000 −1.30986
$$395$$ 12.0000 0.603786
$$396$$ −4.00000 −0.201008
$$397$$ 2.00000 0.100377 0.0501886 0.998740i $$-0.484018\pi$$
0.0501886 + 0.998740i $$0.484018\pi$$
$$398$$ 12.0000 0.601506
$$399$$ 0 0
$$400$$ 1.00000 0.0500000
$$401$$ 6.00000 0.299626 0.149813 0.988714i $$-0.452133\pi$$
0.149813 + 0.988714i $$0.452133\pi$$
$$402$$ −4.00000 −0.199502
$$403$$ 48.0000 2.39105
$$404$$ 18.0000 0.895533
$$405$$ −1.00000 −0.0496904
$$406$$ 0 0
$$407$$ −24.0000 −1.18964
$$408$$ 6.00000 0.297044
$$409$$ 10.0000 0.494468 0.247234 0.968956i $$-0.420478\pi$$
0.247234 + 0.968956i $$0.420478\pi$$
$$410$$ 10.0000 0.493865
$$411$$ 2.00000 0.0986527
$$412$$ −16.0000 −0.788263
$$413$$ 0 0
$$414$$ −1.00000 −0.0491473
$$415$$ 16.0000 0.785409
$$416$$ 6.00000 0.294174
$$417$$ 12.0000 0.587643
$$418$$ 16.0000 0.782586
$$419$$ −12.0000 −0.586238 −0.293119 0.956076i $$-0.594693\pi$$
−0.293119 + 0.956076i $$0.594693\pi$$
$$420$$ 0 0
$$421$$ −38.0000 −1.85201 −0.926003 0.377515i $$-0.876779\pi$$
−0.926003 + 0.377515i $$0.876779\pi$$
$$422$$ 20.0000 0.973585
$$423$$ −8.00000 −0.388973
$$424$$ 14.0000 0.679900
$$425$$ −6.00000 −0.291043
$$426$$ −8.00000 −0.387601
$$427$$ 0 0
$$428$$ 8.00000 0.386695
$$429$$ 24.0000 1.15873
$$430$$ 4.00000 0.192897
$$431$$ 8.00000 0.385346 0.192673 0.981263i $$-0.438284\pi$$
0.192673 + 0.981263i $$0.438284\pi$$
$$432$$ 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
$$435$$ 6.00000 0.287678
$$436$$ 2.00000 0.0957826
$$437$$ 4.00000 0.191346
$$438$$ −2.00000 −0.0955637
$$439$$ −24.0000 −1.14546 −0.572729 0.819745i $$-0.694115\pi$$
−0.572729 + 0.819745i $$0.694115\pi$$
$$440$$ −4.00000 −0.190693
$$441$$ −7.00000 −0.333333
$$442$$ −36.0000 −1.71235
$$443$$ 12.0000 0.570137 0.285069 0.958507i $$-0.407984\pi$$
0.285069 + 0.958507i $$0.407984\pi$$
$$444$$ 6.00000 0.284747
$$445$$ 2.00000 0.0948091
$$446$$ 20.0000 0.947027
$$447$$ 10.0000 0.472984
$$448$$ 0 0
$$449$$ 10.0000 0.471929 0.235965 0.971762i $$-0.424175\pi$$
0.235965 + 0.971762i $$0.424175\pi$$
$$450$$ −1.00000 −0.0471405
$$451$$ −40.0000 −1.88353
$$452$$ −14.0000 −0.658505
$$453$$ 16.0000 0.751746
$$454$$ −8.00000 −0.375459
$$455$$ 0 0
$$456$$ −4.00000 −0.187317
$$457$$ 10.0000 0.467780 0.233890 0.972263i $$-0.424854\pi$$
0.233890 + 0.972263i $$0.424854\pi$$
$$458$$ 14.0000 0.654177
$$459$$ −6.00000 −0.280056
$$460$$ −1.00000 −0.0466252
$$461$$ 2.00000 0.0931493 0.0465746 0.998915i $$-0.485169\pi$$
0.0465746 + 0.998915i $$0.485169\pi$$
$$462$$ 0 0
$$463$$ −20.0000 −0.929479 −0.464739 0.885448i $$-0.653852\pi$$
−0.464739 + 0.885448i $$0.653852\pi$$
$$464$$ −6.00000 −0.278543
$$465$$ 8.00000 0.370991
$$466$$ −22.0000 −1.01913
$$467$$ 8.00000 0.370196 0.185098 0.982720i $$-0.440740\pi$$
0.185098 + 0.982720i $$0.440740\pi$$
$$468$$ −6.00000 −0.277350
$$469$$ 0 0
$$470$$ −8.00000 −0.369012
$$471$$ 6.00000 0.276465
$$472$$ 0 0
$$473$$ −16.0000 −0.735681
$$474$$ 12.0000 0.551178
$$475$$ 4.00000 0.183533
$$476$$ 0 0
$$477$$ −14.0000 −0.641016
$$478$$ 8.00000 0.365911
$$479$$ 24.0000 1.09659 0.548294 0.836286i $$-0.315277\pi$$
0.548294 + 0.836286i $$0.315277\pi$$
$$480$$ 1.00000 0.0456435
$$481$$ −36.0000 −1.64146
$$482$$ −2.00000 −0.0910975
$$483$$ 0 0
$$484$$ 5.00000 0.227273
$$485$$ 14.0000 0.635707
$$486$$ −1.00000 −0.0453609
$$487$$ 20.0000 0.906287 0.453143 0.891438i $$-0.350303\pi$$
0.453143 + 0.891438i $$0.350303\pi$$
$$488$$ −10.0000 −0.452679
$$489$$ 4.00000 0.180886
$$490$$ −7.00000 −0.316228
$$491$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$492$$ 10.0000 0.450835
$$493$$ 36.0000 1.62136
$$494$$ 24.0000 1.07981
$$495$$ 4.00000 0.179787
$$496$$ −8.00000 −0.359211
$$497$$ 0 0
$$498$$ 16.0000 0.716977
$$499$$ 12.0000 0.537194 0.268597 0.963253i $$-0.413440\pi$$
0.268597 + 0.963253i $$0.413440\pi$$
$$500$$ −1.00000 −0.0447214
$$501$$ −16.0000 −0.714827
$$502$$ 12.0000 0.535586
$$503$$ −24.0000 −1.07011 −0.535054 0.844818i $$-0.679709\pi$$
−0.535054 + 0.844818i $$0.679709\pi$$
$$504$$ 0 0
$$505$$ −18.0000 −0.800989
$$506$$ 4.00000 0.177822
$$507$$ 23.0000 1.02147
$$508$$ 12.0000 0.532414
$$509$$ −6.00000 −0.265945 −0.132973 0.991120i $$-0.542452\pi$$
−0.132973 + 0.991120i $$0.542452\pi$$
$$510$$ −6.00000 −0.265684
$$511$$ 0 0
$$512$$ −1.00000 −0.0441942
$$513$$ 4.00000 0.176604
$$514$$ −6.00000 −0.264649
$$515$$ 16.0000 0.705044
$$516$$ 4.00000 0.176090
$$517$$ 32.0000 1.40736
$$518$$ 0 0
$$519$$ 18.0000 0.790112
$$520$$ −6.00000 −0.263117
$$521$$ −42.0000 −1.84005 −0.920027 0.391856i $$-0.871833\pi$$
−0.920027 + 0.391856i $$0.871833\pi$$
$$522$$ 6.00000 0.262613
$$523$$ −20.0000 −0.874539 −0.437269 0.899331i $$-0.644054\pi$$
−0.437269 + 0.899331i $$0.644054\pi$$
$$524$$ 8.00000 0.349482
$$525$$ 0 0
$$526$$ 32.0000 1.39527
$$527$$ 48.0000 2.09091
$$528$$ −4.00000 −0.174078
$$529$$ 1.00000 0.0434783
$$530$$ −14.0000 −0.608121
$$531$$ 0 0
$$532$$ 0 0
$$533$$ −60.0000 −2.59889
$$534$$ 2.00000 0.0865485
$$535$$ −8.00000 −0.345870
$$536$$ −4.00000 −0.172774
$$537$$ 0 0
$$538$$ −10.0000 −0.431131
$$539$$ 28.0000 1.20605
$$540$$ −1.00000 −0.0430331
$$541$$ −34.0000 −1.46177 −0.730887 0.682498i $$-0.760893\pi$$
−0.730887 + 0.682498i $$0.760893\pi$$
$$542$$ 8.00000 0.343629
$$543$$ −6.00000 −0.257485
$$544$$ 6.00000 0.257248
$$545$$ −2.00000 −0.0856706
$$546$$ 0 0
$$547$$ 20.0000 0.855138 0.427569 0.903983i $$-0.359370\pi$$
0.427569 + 0.903983i $$0.359370\pi$$
$$548$$ 2.00000 0.0854358
$$549$$ 10.0000 0.426790
$$550$$ 4.00000 0.170561
$$551$$ −24.0000 −1.02243
$$552$$ −1.00000 −0.0425628
$$553$$ 0 0
$$554$$ 30.0000 1.27458
$$555$$ −6.00000 −0.254686
$$556$$ 12.0000 0.508913
$$557$$ 18.0000 0.762684 0.381342 0.924434i $$-0.375462\pi$$
0.381342 + 0.924434i $$0.375462\pi$$
$$558$$ 8.00000 0.338667
$$559$$ −24.0000 −1.01509
$$560$$ 0 0
$$561$$ 24.0000 1.01328
$$562$$ −30.0000 −1.26547
$$563$$ −8.00000 −0.337160 −0.168580 0.985688i $$-0.553918\pi$$
−0.168580 + 0.985688i $$0.553918\pi$$
$$564$$ −8.00000 −0.336861
$$565$$ 14.0000 0.588984
$$566$$ −4.00000 −0.168133
$$567$$ 0 0
$$568$$ −8.00000 −0.335673
$$569$$ −18.0000 −0.754599 −0.377300 0.926091i $$-0.623147\pi$$
−0.377300 + 0.926091i $$0.623147\pi$$
$$570$$ 4.00000 0.167542
$$571$$ 20.0000 0.836974 0.418487 0.908223i $$-0.362561\pi$$
0.418487 + 0.908223i $$0.362561\pi$$
$$572$$ 24.0000 1.00349
$$573$$ −8.00000 −0.334205
$$574$$ 0 0
$$575$$ 1.00000 0.0417029
$$576$$ 1.00000 0.0416667
$$577$$ 34.0000 1.41544 0.707719 0.706494i $$-0.249724\pi$$
0.707719 + 0.706494i $$0.249724\pi$$
$$578$$ −19.0000 −0.790296
$$579$$ 10.0000 0.415586
$$580$$ 6.00000 0.249136
$$581$$ 0 0
$$582$$ 14.0000 0.580319
$$583$$ 56.0000 2.31928
$$584$$ −2.00000 −0.0827606
$$585$$ 6.00000 0.248069
$$586$$ 14.0000 0.578335
$$587$$ −4.00000 −0.165098 −0.0825488 0.996587i $$-0.526306\pi$$
−0.0825488 + 0.996587i $$0.526306\pi$$
$$588$$ −7.00000 −0.288675
$$589$$ −32.0000 −1.31854
$$590$$ 0 0
$$591$$ 26.0000 1.06950
$$592$$ 6.00000 0.246598
$$593$$ 6.00000 0.246390 0.123195 0.992382i $$-0.460686\pi$$
0.123195 + 0.992382i $$0.460686\pi$$
$$594$$ 4.00000 0.164122
$$595$$ 0 0
$$596$$ 10.0000 0.409616
$$597$$ −12.0000 −0.491127
$$598$$ 6.00000 0.245358
$$599$$ −40.0000 −1.63436 −0.817178 0.576386i $$-0.804463\pi$$
−0.817178 + 0.576386i $$0.804463\pi$$
$$600$$ −1.00000 −0.0408248
$$601$$ 10.0000 0.407909 0.203954 0.978980i $$-0.434621\pi$$
0.203954 + 0.978980i $$0.434621\pi$$
$$602$$ 0 0
$$603$$ 4.00000 0.162893
$$604$$ 16.0000 0.651031
$$605$$ −5.00000 −0.203279
$$606$$ −18.0000 −0.731200
$$607$$ −28.0000 −1.13648 −0.568242 0.822861i $$-0.692376\pi$$
−0.568242 + 0.822861i $$0.692376\pi$$
$$608$$ −4.00000 −0.162221
$$609$$ 0 0
$$610$$ 10.0000 0.404888
$$611$$ 48.0000 1.94187
$$612$$ −6.00000 −0.242536
$$613$$ 14.0000 0.565455 0.282727 0.959200i $$-0.408761\pi$$
0.282727 + 0.959200i $$0.408761\pi$$
$$614$$ −12.0000 −0.484281
$$615$$ −10.0000 −0.403239
$$616$$ 0 0
$$617$$ 18.0000 0.724653 0.362326 0.932051i $$-0.381983\pi$$
0.362326 + 0.932051i $$0.381983\pi$$
$$618$$ 16.0000 0.643614
$$619$$ 20.0000 0.803868 0.401934 0.915669i $$-0.368338\pi$$
0.401934 + 0.915669i $$0.368338\pi$$
$$620$$ 8.00000 0.321288
$$621$$ 1.00000 0.0401286
$$622$$ 8.00000 0.320771
$$623$$ 0 0
$$624$$ −6.00000 −0.240192
$$625$$ 1.00000 0.0400000
$$626$$ −10.0000 −0.399680
$$627$$ −16.0000 −0.638978
$$628$$ 6.00000 0.239426
$$629$$ −36.0000 −1.43541
$$630$$ 0 0
$$631$$ 20.0000 0.796187 0.398094 0.917345i $$-0.369672\pi$$
0.398094 + 0.917345i $$0.369672\pi$$
$$632$$ 12.0000 0.477334
$$633$$ −20.0000 −0.794929
$$634$$ −18.0000 −0.714871
$$635$$ −12.0000 −0.476205
$$636$$ −14.0000 −0.555136
$$637$$ 42.0000 1.66410
$$638$$ −24.0000 −0.950169
$$639$$ 8.00000 0.316475
$$640$$ 1.00000 0.0395285
$$641$$ −2.00000 −0.0789953 −0.0394976 0.999220i $$-0.512576\pi$$
−0.0394976 + 0.999220i $$0.512576\pi$$
$$642$$ −8.00000 −0.315735
$$643$$ −44.0000 −1.73519 −0.867595 0.497271i $$-0.834335\pi$$
−0.867595 + 0.497271i $$0.834335\pi$$
$$644$$ 0 0
$$645$$ −4.00000 −0.157500
$$646$$ 24.0000 0.944267
$$647$$ 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
$$652$$ 4.00000 0.156652
$$653$$ −14.0000 −0.547862 −0.273931 0.961749i $$-0.588324\pi$$
−0.273931 + 0.961749i $$0.588324\pi$$
$$654$$ −2.00000 −0.0782062
$$655$$ −8.00000 −0.312586
$$656$$ 10.0000 0.390434
$$657$$ 2.00000 0.0780274
$$658$$ 0 0
$$659$$ −36.0000 −1.40236 −0.701180 0.712984i $$-0.747343\pi$$
−0.701180 + 0.712984i $$0.747343\pi$$
$$660$$ 4.00000 0.155700
$$661$$ 34.0000 1.32245 0.661223 0.750189i $$-0.270038\pi$$
0.661223 + 0.750189i $$0.270038\pi$$
$$662$$ 20.0000 0.777322
$$663$$ 36.0000 1.39812
$$664$$ 16.0000 0.620920
$$665$$ 0 0
$$666$$ −6.00000 −0.232495
$$667$$ −6.00000 −0.232321
$$668$$ −16.0000 −0.619059
$$669$$ −20.0000 −0.773245
$$670$$ 4.00000 0.154533
$$671$$ −40.0000 −1.54418
$$672$$ 0 0
$$673$$ −22.0000 −0.848038 −0.424019 0.905653i $$-0.639381\pi$$
−0.424019 + 0.905653i $$0.639381\pi$$
$$674$$ 14.0000 0.539260
$$675$$ 1.00000 0.0384900
$$676$$ 23.0000 0.884615
$$677$$ −6.00000 −0.230599 −0.115299 0.993331i $$-0.536783\pi$$
−0.115299 + 0.993331i $$0.536783\pi$$
$$678$$ 14.0000 0.537667
$$679$$ 0 0
$$680$$ −6.00000 −0.230089
$$681$$ 8.00000 0.306561
$$682$$ −32.0000 −1.22534
$$683$$ 44.0000 1.68361 0.841807 0.539779i $$-0.181492\pi$$
0.841807 + 0.539779i $$0.181492\pi$$
$$684$$ 4.00000 0.152944
$$685$$ −2.00000 −0.0764161
$$686$$ 0 0
$$687$$ −14.0000 −0.534133
$$688$$ 4.00000 0.152499
$$689$$ 84.0000 3.20015
$$690$$ 1.00000 0.0380693
$$691$$ −28.0000 −1.06517 −0.532585 0.846376i $$-0.678779\pi$$
−0.532585 + 0.846376i $$0.678779\pi$$
$$692$$ 18.0000 0.684257
$$693$$ 0 0
$$694$$ 28.0000 1.06287
$$695$$ −12.0000 −0.455186
$$696$$ 6.00000 0.227429
$$697$$ −60.0000 −2.27266
$$698$$ 34.0000 1.28692
$$699$$ 22.0000 0.832116
$$700$$ 0 0
$$701$$ 18.0000 0.679851 0.339925 0.940452i $$-0.389598\pi$$
0.339925 + 0.940452i $$0.389598\pi$$
$$702$$ 6.00000 0.226455
$$703$$ 24.0000 0.905177
$$704$$ −4.00000 −0.150756
$$705$$ 8.00000 0.301297
$$706$$ −6.00000 −0.225813
$$707$$ 0 0
$$708$$ 0 0
$$709$$ 10.0000 0.375558 0.187779 0.982211i $$-0.439871\pi$$
0.187779 + 0.982211i $$0.439871\pi$$
$$710$$ 8.00000 0.300235
$$711$$ −12.0000 −0.450035
$$712$$ 2.00000 0.0749532
$$713$$ −8.00000 −0.299602
$$714$$ 0 0
$$715$$ −24.0000 −0.897549
$$716$$ 0 0
$$717$$ −8.00000 −0.298765
$$718$$ 8.00000 0.298557
$$719$$ −40.0000 −1.49175 −0.745874 0.666087i $$-0.767968\pi$$
−0.745874 + 0.666087i $$0.767968\pi$$
$$720$$ −1.00000 −0.0372678
$$721$$ 0 0
$$722$$ 3.00000 0.111648
$$723$$ 2.00000 0.0743808
$$724$$ −6.00000 −0.222988
$$725$$ −6.00000 −0.222834
$$726$$ −5.00000 −0.185567
$$727$$ 8.00000 0.296704 0.148352 0.988935i $$-0.452603\pi$$
0.148352 + 0.988935i $$0.452603\pi$$
$$728$$ 0 0
$$729$$ 1.00000 0.0370370
$$730$$ 2.00000 0.0740233
$$731$$ −24.0000 −0.887672
$$732$$ 10.0000 0.369611
$$733$$ 6.00000 0.221615 0.110808 0.993842i $$-0.464656\pi$$
0.110808 + 0.993842i $$0.464656\pi$$
$$734$$ 32.0000 1.18114
$$735$$ 7.00000 0.258199
$$736$$ −1.00000 −0.0368605
$$737$$ −16.0000 −0.589368
$$738$$ −10.0000 −0.368105
$$739$$ −12.0000 −0.441427 −0.220714 0.975339i $$-0.570839\pi$$
−0.220714 + 0.975339i $$0.570839\pi$$
$$740$$ −6.00000 −0.220564
$$741$$ −24.0000 −0.881662
$$742$$ 0 0
$$743$$ −24.0000 −0.880475 −0.440237 0.897881i $$-0.645106\pi$$
−0.440237 + 0.897881i $$0.645106\pi$$
$$744$$ 8.00000 0.293294
$$745$$ −10.0000 −0.366372
$$746$$ 18.0000 0.659027
$$747$$ −16.0000 −0.585409
$$748$$ 24.0000 0.877527
$$749$$ 0 0
$$750$$ 1.00000 0.0365148
$$751$$ −28.0000 −1.02173 −0.510867 0.859660i $$-0.670676\pi$$
−0.510867 + 0.859660i $$0.670676\pi$$
$$752$$ −8.00000 −0.291730
$$753$$ −12.0000 −0.437304
$$754$$ −36.0000 −1.31104
$$755$$ −16.0000 −0.582300
$$756$$ 0 0
$$757$$ 54.0000 1.96266 0.981332 0.192323i $$-0.0616021\pi$$
0.981332 + 0.192323i $$0.0616021\pi$$
$$758$$ −4.00000 −0.145287
$$759$$ −4.00000 −0.145191
$$760$$ 4.00000 0.145095
$$761$$ 10.0000 0.362500 0.181250 0.983437i $$-0.441986\pi$$
0.181250 + 0.983437i $$0.441986\pi$$
$$762$$ −12.0000 −0.434714
$$763$$ 0 0
$$764$$ −8.00000 −0.289430
$$765$$ 6.00000 0.216930
$$766$$ −16.0000 −0.578103
$$767$$ 0 0
$$768$$ 1.00000 0.0360844
$$769$$ −6.00000 −0.216366 −0.108183 0.994131i $$-0.534503\pi$$
−0.108183 + 0.994131i $$0.534503\pi$$
$$770$$ 0 0
$$771$$ 6.00000 0.216085
$$772$$ 10.0000 0.359908
$$773$$ 2.00000 0.0719350 0.0359675 0.999353i $$-0.488549\pi$$
0.0359675 + 0.999353i $$0.488549\pi$$
$$774$$ −4.00000 −0.143777
$$775$$ −8.00000 −0.287368
$$776$$ 14.0000 0.502571
$$777$$ 0 0
$$778$$ 6.00000 0.215110
$$779$$ 40.0000 1.43315
$$780$$ 6.00000 0.214834
$$781$$ −32.0000 −1.14505
$$782$$ 6.00000 0.214560
$$783$$ −6.00000 −0.214423
$$784$$ −7.00000 −0.250000
$$785$$ −6.00000 −0.214149
$$786$$ −8.00000 −0.285351
$$787$$ 36.0000 1.28326 0.641631 0.767014i $$-0.278258\pi$$
0.641631 + 0.767014i $$0.278258\pi$$
$$788$$ 26.0000 0.926212
$$789$$ −32.0000 −1.13923
$$790$$ −12.0000 −0.426941
$$791$$ 0 0
$$792$$ 4.00000 0.142134
$$793$$ −60.0000 −2.13066
$$794$$ −2.00000 −0.0709773
$$795$$ 14.0000 0.496529
$$796$$ −12.0000 −0.425329
$$797$$ 18.0000 0.637593 0.318796 0.947823i $$-0.396721\pi$$
0.318796 + 0.947823i $$0.396721\pi$$
$$798$$ 0 0
$$799$$ 48.0000 1.69812
$$800$$ −1.00000 −0.0353553
$$801$$ −2.00000 −0.0706665
$$802$$ −6.00000 −0.211867
$$803$$ −8.00000 −0.282314
$$804$$ 4.00000 0.141069
$$805$$ 0 0
$$806$$ −48.0000 −1.69073
$$807$$ 10.0000 0.352017
$$808$$ −18.0000 −0.633238
$$809$$ −30.0000 −1.05474 −0.527372 0.849635i $$-0.676823\pi$$
−0.527372 + 0.849635i $$0.676823\pi$$
$$810$$ 1.00000 0.0351364
$$811$$ 20.0000 0.702295 0.351147 0.936320i $$-0.385792\pi$$
0.351147 + 0.936320i $$0.385792\pi$$
$$812$$ 0 0
$$813$$ −8.00000 −0.280572
$$814$$ 24.0000 0.841200
$$815$$ −4.00000 −0.140114
$$816$$ −6.00000 −0.210042
$$817$$ 16.0000 0.559769
$$818$$ −10.0000 −0.349642
$$819$$ 0 0
$$820$$ −10.0000 −0.349215
$$821$$ −46.0000 −1.60541 −0.802706 0.596376i $$-0.796607\pi$$
−0.802706 + 0.596376i $$0.796607\pi$$
$$822$$ −2.00000 −0.0697580
$$823$$ 20.0000 0.697156 0.348578 0.937280i $$-0.386665\pi$$
0.348578 + 0.937280i $$0.386665\pi$$
$$824$$ 16.0000 0.557386
$$825$$ −4.00000 −0.139262
$$826$$ 0 0
$$827$$ −8.00000 −0.278187 −0.139094 0.990279i $$-0.544419\pi$$
−0.139094 + 0.990279i $$0.544419\pi$$
$$828$$ 1.00000 0.0347524
$$829$$ −18.0000 −0.625166 −0.312583 0.949890i $$-0.601194\pi$$
−0.312583 + 0.949890i $$0.601194\pi$$
$$830$$ −16.0000 −0.555368
$$831$$ −30.0000 −1.04069
$$832$$ −6.00000 −0.208013
$$833$$ 42.0000 1.45521
$$834$$ −12.0000 −0.415526
$$835$$ 16.0000 0.553703
$$836$$ −16.0000 −0.553372
$$837$$ −8.00000 −0.276520
$$838$$ 12.0000 0.414533
$$839$$ −48.0000 −1.65714 −0.828572 0.559883i $$-0.810846\pi$$
−0.828572 + 0.559883i $$0.810846\pi$$
$$840$$ 0 0
$$841$$ 7.00000 0.241379
$$842$$ 38.0000 1.30957
$$843$$ 30.0000 1.03325
$$844$$ −20.0000 −0.688428
$$845$$ −23.0000 −0.791224
$$846$$ 8.00000 0.275046
$$847$$ 0 0
$$848$$ −14.0000 −0.480762
$$849$$ 4.00000 0.137280
$$850$$ 6.00000 0.205798
$$851$$ 6.00000 0.205677
$$852$$ 8.00000 0.274075
$$853$$ −22.0000 −0.753266 −0.376633 0.926363i $$-0.622918\pi$$
−0.376633 + 0.926363i $$0.622918\pi$$
$$854$$ 0 0
$$855$$ −4.00000 −0.136797
$$856$$ −8.00000 −0.273434
$$857$$ −42.0000 −1.43469 −0.717346 0.696717i $$-0.754643\pi$$
−0.717346 + 0.696717i $$0.754643\pi$$
$$858$$ −24.0000 −0.819346
$$859$$ 4.00000 0.136478 0.0682391 0.997669i $$-0.478262\pi$$
0.0682391 + 0.997669i $$0.478262\pi$$
$$860$$ −4.00000 −0.136399
$$861$$ 0 0
$$862$$ −8.00000 −0.272481
$$863$$ 24.0000 0.816970 0.408485 0.912765i $$-0.366057\pi$$
0.408485 + 0.912765i $$0.366057\pi$$
$$864$$ −1.00000 −0.0340207
$$865$$ −18.0000 −0.612018
$$866$$ −2.00000 −0.0679628
$$867$$ 19.0000 0.645274
$$868$$ 0 0
$$869$$ 48.0000 1.62829
$$870$$ −6.00000 −0.203419
$$871$$ −24.0000 −0.813209
$$872$$ −2.00000 −0.0677285
$$873$$ −14.0000 −0.473828
$$874$$ −4.00000 −0.135302
$$875$$ 0 0
$$876$$ 2.00000 0.0675737
$$877$$ 2.00000 0.0675352 0.0337676 0.999430i $$-0.489249\pi$$
0.0337676 + 0.999430i $$0.489249\pi$$
$$878$$ 24.0000 0.809961
$$879$$ −14.0000 −0.472208
$$880$$ 4.00000 0.134840
$$881$$ 6.00000 0.202145 0.101073 0.994879i $$-0.467773\pi$$
0.101073 + 0.994879i $$0.467773\pi$$
$$882$$ 7.00000 0.235702
$$883$$ 44.0000 1.48072 0.740359 0.672212i $$-0.234656\pi$$
0.740359 + 0.672212i $$0.234656\pi$$
$$884$$ 36.0000 1.21081
$$885$$ 0 0
$$886$$ −12.0000 −0.403148
$$887$$ 8.00000 0.268614 0.134307 0.990940i $$-0.457119\pi$$
0.134307 + 0.990940i $$0.457119\pi$$
$$888$$ −6.00000 −0.201347
$$889$$ 0 0
$$890$$ −2.00000 −0.0670402
$$891$$ −4.00000 −0.134005
$$892$$ −20.0000 −0.669650
$$893$$ −32.0000 −1.07084
$$894$$ −10.0000 −0.334450
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −6.00000 −0.200334
$$898$$ −10.0000 −0.333704
$$899$$ 48.0000 1.60089
$$900$$ 1.00000 0.0333333
$$901$$ 84.0000 2.79845
$$902$$ 40.0000 1.33185
$$903$$ 0 0
$$904$$ 14.0000 0.465633
$$905$$ 6.00000 0.199447
$$906$$ −16.0000 −0.531564
$$907$$ −4.00000 −0.132818 −0.0664089 0.997792i $$-0.521154\pi$$
−0.0664089 + 0.997792i $$0.521154\pi$$
$$908$$ 8.00000 0.265489
$$909$$ 18.0000 0.597022
$$910$$ 0 0
$$911$$ 8.00000 0.265052 0.132526 0.991180i $$-0.457691\pi$$
0.132526 + 0.991180i $$0.457691\pi$$
$$912$$ 4.00000 0.132453
$$913$$ 64.0000 2.11809
$$914$$ −10.0000 −0.330771
$$915$$ −10.0000 −0.330590
$$916$$ −14.0000 −0.462573
$$917$$ 0 0
$$918$$ 6.00000 0.198030
$$919$$ 12.0000 0.395843 0.197922 0.980218i $$-0.436581\pi$$
0.197922 + 0.980218i $$0.436581\pi$$
$$920$$ 1.00000 0.0329690
$$921$$ 12.0000 0.395413
$$922$$ −2.00000 −0.0658665
$$923$$ −48.0000 −1.57994
$$924$$ 0 0
$$925$$ 6.00000 0.197279
$$926$$ 20.0000 0.657241
$$927$$ −16.0000 −0.525509
$$928$$ 6.00000 0.196960
$$929$$ −6.00000 −0.196854 −0.0984268 0.995144i $$-0.531381\pi$$
−0.0984268 + 0.995144i $$0.531381\pi$$
$$930$$ −8.00000 −0.262330
$$931$$ −28.0000 −0.917663
$$932$$ 22.0000 0.720634
$$933$$ −8.00000 −0.261908
$$934$$ −8.00000 −0.261768
$$935$$ −24.0000 −0.784884
$$936$$ 6.00000 0.196116
$$937$$ 10.0000 0.326686 0.163343 0.986569i $$-0.447772\pi$$
0.163343 + 0.986569i $$0.447772\pi$$
$$938$$ 0 0
$$939$$ 10.0000 0.326338
$$940$$ 8.00000 0.260931
$$941$$ −14.0000 −0.456387 −0.228193 0.973616i $$-0.573282\pi$$
−0.228193 + 0.973616i $$0.573282\pi$$
$$942$$ −6.00000 −0.195491
$$943$$ 10.0000 0.325645
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 16.0000 0.520205
$$947$$ −20.0000 −0.649913 −0.324956 0.945729i $$-0.605350\pi$$
−0.324956 + 0.945729i $$0.605350\pi$$
$$948$$ −12.0000 −0.389742
$$949$$ −12.0000 −0.389536
$$950$$ −4.00000 −0.129777
$$951$$ 18.0000 0.583690
$$952$$ 0 0
$$953$$ −22.0000 −0.712650 −0.356325 0.934362i $$-0.615970\pi$$
−0.356325 + 0.934362i $$0.615970\pi$$
$$954$$ 14.0000 0.453267
$$955$$ 8.00000 0.258874
$$956$$ −8.00000 −0.258738
$$957$$ 24.0000 0.775810
$$958$$ −24.0000 −0.775405
$$959$$ 0 0
$$960$$ −1.00000 −0.0322749
$$961$$ 33.0000 1.06452
$$962$$ 36.0000 1.16069
$$963$$ 8.00000 0.257796
$$964$$ 2.00000 0.0644157
$$965$$ −10.0000 −0.321911
$$966$$ 0 0
$$967$$ 28.0000 0.900419 0.450210 0.892923i $$-0.351349\pi$$
0.450210 + 0.892923i $$0.351349\pi$$
$$968$$ −5.00000 −0.160706
$$969$$ −24.0000 −0.770991
$$970$$ −14.0000 −0.449513
$$971$$ 12.0000 0.385098 0.192549 0.981287i $$-0.438325\pi$$
0.192549 + 0.981287i $$0.438325\pi$$
$$972$$ 1.00000 0.0320750
$$973$$ 0 0
$$974$$ −20.0000 −0.640841
$$975$$ −6.00000 −0.192154
$$976$$ 10.0000 0.320092
$$977$$ 2.00000 0.0639857 0.0319928 0.999488i $$-0.489815\pi$$
0.0319928 + 0.999488i $$0.489815\pi$$
$$978$$ −4.00000 −0.127906
$$979$$ 8.00000 0.255681
$$980$$ 7.00000 0.223607
$$981$$ 2.00000 0.0638551
$$982$$ 0 0
$$983$$ −24.0000 −0.765481 −0.382741 0.923856i $$-0.625020\pi$$
−0.382741 + 0.923856i $$0.625020\pi$$
$$984$$ −10.0000 −0.318788
$$985$$ −26.0000 −0.828429
$$986$$ −36.0000 −1.14647
$$987$$ 0 0
$$988$$ −24.0000 −0.763542
$$989$$ 4.00000 0.127193
$$990$$ −4.00000 −0.127128
$$991$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$992$$ 8.00000 0.254000
$$993$$ −20.0000 −0.634681
$$994$$ 0 0
$$995$$ 12.0000 0.380426
$$996$$ −16.0000 −0.506979
$$997$$ −22.0000 −0.696747 −0.348373 0.937356i $$-0.613266\pi$$
−0.348373 + 0.937356i $$0.613266\pi$$
$$998$$ −12.0000 −0.379853
$$999$$ 6.00000 0.189832
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 690.2.a.e.1.1 1
3.2 odd 2 2070.2.a.r.1.1 1
4.3 odd 2 5520.2.a.f.1.1 1
5.2 odd 4 3450.2.d.a.2899.1 2
5.3 odd 4 3450.2.d.a.2899.2 2
5.4 even 2 3450.2.a.p.1.1 1

By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.a.e.1.1 1 1.1 even 1 trivial
2070.2.a.r.1.1 1 3.2 odd 2
3450.2.a.p.1.1 1 5.4 even 2
3450.2.d.a.2899.1 2 5.2 odd 4
3450.2.d.a.2899.2 2 5.3 odd 4
5520.2.a.f.1.1 1 4.3 odd 2