# Properties

 Label 625.2.b.a.624.2 Level $625$ Weight $2$ Character 625.624 Analytic conductor $4.991$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Learn more

## Newspace parameters

 Level: $$N$$ $$=$$ $$625 = 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 625.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$4.99065012633$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{5})$$ Defining polynomial: $$x^{4} + 3 x^{2} + 1$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 25) Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 624.2 Root $$-0.618034i$$ of defining polynomial Character $$\chi$$ $$=$$ 625.624 Dual form 625.2.b.a.624.3

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-0.618034i q^{2} +1.00000i q^{3} +1.61803 q^{4} +0.618034 q^{6} +1.61803i q^{7} -2.23607i q^{8} +2.00000 q^{9} +O(q^{10})$$ $$q-0.618034i q^{2} +1.00000i q^{3} +1.61803 q^{4} +0.618034 q^{6} +1.61803i q^{7} -2.23607i q^{8} +2.00000 q^{9} -0.763932 q^{11} +1.61803i q^{12} -4.85410i q^{13} +1.00000 q^{14} +1.85410 q^{16} +0.763932i q^{17} -1.23607i q^{18} +5.85410 q^{19} -1.61803 q^{21} +0.472136i q^{22} +8.23607i q^{23} +2.23607 q^{24} -3.00000 q^{26} +5.00000i q^{27} +2.61803i q^{28} +1.38197 q^{29} -3.00000 q^{31} -5.61803i q^{32} -0.763932i q^{33} +0.472136 q^{34} +3.23607 q^{36} -4.23607i q^{37} -3.61803i q^{38} +4.85410 q^{39} -5.23607 q^{41} +1.00000i q^{42} +1.85410i q^{43} -1.23607 q^{44} +5.09017 q^{46} +1.61803i q^{47} +1.85410i q^{48} +4.38197 q^{49} -0.763932 q^{51} -7.85410i q^{52} +5.47214i q^{53} +3.09017 q^{54} +3.61803 q^{56} +5.85410i q^{57} -0.854102i q^{58} +4.14590 q^{59} -4.70820 q^{61} +1.85410i q^{62} +3.23607i q^{63} +0.236068 q^{64} -0.472136 q^{66} -9.23607i q^{67} +1.23607i q^{68} -8.23607 q^{69} -4.38197 q^{71} -4.47214i q^{72} -9.00000i q^{73} -2.61803 q^{74} +9.47214 q^{76} -1.23607i q^{77} -3.00000i q^{78} -3.09017 q^{79} +1.00000 q^{81} +3.23607i q^{82} -1.76393i q^{83} -2.61803 q^{84} +1.14590 q^{86} +1.38197i q^{87} +1.70820i q^{88} -8.94427 q^{89} +7.85410 q^{91} +13.3262i q^{92} -3.00000i q^{93} +1.00000 q^{94} +5.61803 q^{96} -2.85410i q^{97} -2.70820i q^{98} -1.52786 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4} - 2 q^{6} + 8 q^{9} + O(q^{10})$$ $$4 q + 2 q^{4} - 2 q^{6} + 8 q^{9} - 12 q^{11} + 4 q^{14} - 6 q^{16} + 10 q^{19} - 2 q^{21} - 12 q^{26} + 10 q^{29} - 12 q^{31} - 16 q^{34} + 4 q^{36} + 6 q^{39} - 12 q^{41} + 4 q^{44} - 2 q^{46} + 22 q^{49} - 12 q^{51} - 10 q^{54} + 10 q^{56} + 30 q^{59} + 8 q^{61} - 8 q^{64} + 16 q^{66} - 24 q^{69} - 22 q^{71} - 6 q^{74} + 20 q^{76} + 10 q^{79} + 4 q^{81} - 6 q^{84} + 18 q^{86} + 18 q^{91} + 4 q^{94} + 18 q^{96} - 24 q^{99} + O(q^{100})$$

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/625\mathbb{Z}\right)^\times$$.

 $$n$$ $$2$$ $$\chi(n)$$ $$-1$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 0.618034i − 0.437016i −0.975835 0.218508i $$-0.929881\pi$$
0.975835 0.218508i $$-0.0701190\pi$$
$$3$$ 1.00000i 0.577350i 0.957427 + 0.288675i $$0.0932147\pi$$
−0.957427 + 0.288675i $$0.906785\pi$$
$$4$$ 1.61803 0.809017
$$5$$ 0 0
$$6$$ 0.618034 0.252311
$$7$$ 1.61803i 0.611559i 0.952102 + 0.305780i $$0.0989171\pi$$
−0.952102 + 0.305780i $$0.901083\pi$$
$$8$$ − 2.23607i − 0.790569i
$$9$$ 2.00000 0.666667
$$10$$ 0 0
$$11$$ −0.763932 −0.230334 −0.115167 0.993346i $$-0.536740\pi$$
−0.115167 + 0.993346i $$0.536740\pi$$
$$12$$ 1.61803i 0.467086i
$$13$$ − 4.85410i − 1.34629i −0.739512 0.673143i $$-0.764944\pi$$
0.739512 0.673143i $$-0.235056\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 0 0
$$16$$ 1.85410 0.463525
$$17$$ 0.763932i 0.185281i 0.995700 + 0.0926404i $$0.0295307\pi$$
−0.995700 + 0.0926404i $$0.970469\pi$$
$$18$$ − 1.23607i − 0.291344i
$$19$$ 5.85410 1.34302 0.671512 0.740994i $$-0.265645\pi$$
0.671512 + 0.740994i $$0.265645\pi$$
$$20$$ 0 0
$$21$$ −1.61803 −0.353084
$$22$$ 0.472136i 0.100660i
$$23$$ 8.23607i 1.71734i 0.512530 + 0.858669i $$0.328708\pi$$
−0.512530 + 0.858669i $$0.671292\pi$$
$$24$$ 2.23607 0.456435
$$25$$ 0 0
$$26$$ −3.00000 −0.588348
$$27$$ 5.00000i 0.962250i
$$28$$ 2.61803i 0.494762i
$$29$$ 1.38197 0.256625 0.128312 0.991734i $$-0.459044\pi$$
0.128312 + 0.991734i $$0.459044\pi$$
$$30$$ 0 0
$$31$$ −3.00000 −0.538816 −0.269408 0.963026i $$-0.586828\pi$$
−0.269408 + 0.963026i $$0.586828\pi$$
$$32$$ − 5.61803i − 0.993137i
$$33$$ − 0.763932i − 0.132983i
$$34$$ 0.472136 0.0809706
$$35$$ 0 0
$$36$$ 3.23607 0.539345
$$37$$ − 4.23607i − 0.696405i −0.937419 0.348203i $$-0.886792\pi$$
0.937419 0.348203i $$-0.113208\pi$$
$$38$$ − 3.61803i − 0.586923i
$$39$$ 4.85410 0.777278
$$40$$ 0 0
$$41$$ −5.23607 −0.817736 −0.408868 0.912593i $$-0.634076\pi$$
−0.408868 + 0.912593i $$0.634076\pi$$
$$42$$ 1.00000i 0.154303i
$$43$$ 1.85410i 0.282748i 0.989956 + 0.141374i $$0.0451520\pi$$
−0.989956 + 0.141374i $$0.954848\pi$$
$$44$$ −1.23607 −0.186344
$$45$$ 0 0
$$46$$ 5.09017 0.750505
$$47$$ 1.61803i 0.236015i 0.993013 + 0.118007i $$0.0376506\pi$$
−0.993013 + 0.118007i $$0.962349\pi$$
$$48$$ 1.85410i 0.267617i
$$49$$ 4.38197 0.625995
$$50$$ 0 0
$$51$$ −0.763932 −0.106972
$$52$$ − 7.85410i − 1.08917i
$$53$$ 5.47214i 0.751656i 0.926690 + 0.375828i $$0.122642\pi$$
−0.926690 + 0.375828i $$0.877358\pi$$
$$54$$ 3.09017 0.420519
$$55$$ 0 0
$$56$$ 3.61803 0.483480
$$57$$ 5.85410i 0.775395i
$$58$$ − 0.854102i − 0.112149i
$$59$$ 4.14590 0.539750 0.269875 0.962895i $$-0.413018\pi$$
0.269875 + 0.962895i $$0.413018\pi$$
$$60$$ 0 0
$$61$$ −4.70820 −0.602824 −0.301412 0.953494i $$-0.597458\pi$$
−0.301412 + 0.953494i $$0.597458\pi$$
$$62$$ 1.85410i 0.235471i
$$63$$ 3.23607i 0.407706i
$$64$$ 0.236068 0.0295085
$$65$$ 0 0
$$66$$ −0.472136 −0.0581159
$$67$$ − 9.23607i − 1.12837i −0.825650 0.564183i $$-0.809191\pi$$
0.825650 0.564183i $$-0.190809\pi$$
$$68$$ 1.23607i 0.149895i
$$69$$ −8.23607 −0.991506
$$70$$ 0 0
$$71$$ −4.38197 −0.520044 −0.260022 0.965603i $$-0.583730\pi$$
−0.260022 + 0.965603i $$0.583730\pi$$
$$72$$ − 4.47214i − 0.527046i
$$73$$ − 9.00000i − 1.05337i −0.850060 0.526685i $$-0.823435\pi$$
0.850060 0.526685i $$-0.176565\pi$$
$$74$$ −2.61803 −0.304340
$$75$$ 0 0
$$76$$ 9.47214 1.08653
$$77$$ − 1.23607i − 0.140863i
$$78$$ − 3.00000i − 0.339683i
$$79$$ −3.09017 −0.347671 −0.173836 0.984775i $$-0.555616\pi$$
−0.173836 + 0.984775i $$0.555616\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 3.23607i 0.357364i
$$83$$ − 1.76393i − 0.193617i −0.995303 0.0968083i $$-0.969137\pi$$
0.995303 0.0968083i $$-0.0308634\pi$$
$$84$$ −2.61803 −0.285651
$$85$$ 0 0
$$86$$ 1.14590 0.123565
$$87$$ 1.38197i 0.148162i
$$88$$ 1.70820i 0.182095i
$$89$$ −8.94427 −0.948091 −0.474045 0.880500i $$-0.657207\pi$$
−0.474045 + 0.880500i $$0.657207\pi$$
$$90$$ 0 0
$$91$$ 7.85410 0.823334
$$92$$ 13.3262i 1.38936i
$$93$$ − 3.00000i − 0.311086i
$$94$$ 1.00000 0.103142
$$95$$ 0 0
$$96$$ 5.61803 0.573388
$$97$$ − 2.85410i − 0.289790i −0.989447 0.144895i $$-0.953716\pi$$
0.989447 0.144895i $$-0.0462845\pi$$
$$98$$ − 2.70820i − 0.273570i
$$99$$ −1.52786 −0.153556
$$100$$ 0 0
$$101$$ −7.47214 −0.743505 −0.371753 0.928332i $$-0.621243\pi$$
−0.371753 + 0.928332i $$0.621243\pi$$
$$102$$ 0.472136i 0.0467484i
$$103$$ − 11.5623i − 1.13927i −0.821899 0.569634i $$-0.807085\pi$$
0.821899 0.569634i $$-0.192915\pi$$
$$104$$ −10.8541 −1.06433
$$105$$ 0 0
$$106$$ 3.38197 0.328486
$$107$$ − 10.4164i − 1.00699i −0.863998 0.503496i $$-0.832047\pi$$
0.863998 0.503496i $$-0.167953\pi$$
$$108$$ 8.09017i 0.778477i
$$109$$ −10.0000 −0.957826 −0.478913 0.877862i $$-0.658969\pi$$
−0.478913 + 0.877862i $$0.658969\pi$$
$$110$$ 0 0
$$111$$ 4.23607 0.402070
$$112$$ 3.00000i 0.283473i
$$113$$ 10.1459i 0.954446i 0.878782 + 0.477223i $$0.158357\pi$$
−0.878782 + 0.477223i $$0.841643\pi$$
$$114$$ 3.61803 0.338860
$$115$$ 0 0
$$116$$ 2.23607 0.207614
$$117$$ − 9.70820i − 0.897524i
$$118$$ − 2.56231i − 0.235879i
$$119$$ −1.23607 −0.113310
$$120$$ 0 0
$$121$$ −10.4164 −0.946946
$$122$$ 2.90983i 0.263444i
$$123$$ − 5.23607i − 0.472120i
$$124$$ −4.85410 −0.435911
$$125$$ 0 0
$$126$$ 2.00000 0.178174
$$127$$ 15.8885i 1.40988i 0.709267 + 0.704940i $$0.249026\pi$$
−0.709267 + 0.704940i $$0.750974\pi$$
$$128$$ − 11.3820i − 1.00603i
$$129$$ −1.85410 −0.163245
$$130$$ 0 0
$$131$$ −17.7984 −1.55505 −0.777526 0.628851i $$-0.783525\pi$$
−0.777526 + 0.628851i $$0.783525\pi$$
$$132$$ − 1.23607i − 0.107586i
$$133$$ 9.47214i 0.821338i
$$134$$ −5.70820 −0.493114
$$135$$ 0 0
$$136$$ 1.70820 0.146477
$$137$$ − 5.94427i − 0.507853i −0.967223 0.253927i $$-0.918278\pi$$
0.967223 0.253927i $$-0.0817222\pi$$
$$138$$ 5.09017i 0.433304i
$$139$$ −5.00000 −0.424094 −0.212047 0.977259i $$-0.568013\pi$$
−0.212047 + 0.977259i $$0.568013\pi$$
$$140$$ 0 0
$$141$$ −1.61803 −0.136263
$$142$$ 2.70820i 0.227267i
$$143$$ 3.70820i 0.310096i
$$144$$ 3.70820 0.309017
$$145$$ 0 0
$$146$$ −5.56231 −0.460340
$$147$$ 4.38197i 0.361418i
$$148$$ − 6.85410i − 0.563404i
$$149$$ −13.9443 −1.14236 −0.571180 0.820825i $$-0.693514\pi$$
−0.571180 + 0.820825i $$0.693514\pi$$
$$150$$ 0 0
$$151$$ −5.56231 −0.452654 −0.226327 0.974051i $$-0.572672\pi$$
−0.226327 + 0.974051i $$0.572672\pi$$
$$152$$ − 13.0902i − 1.06175i
$$153$$ 1.52786i 0.123520i
$$154$$ −0.763932 −0.0615594
$$155$$ 0 0
$$156$$ 7.85410 0.628831
$$157$$ 9.18034i 0.732671i 0.930483 + 0.366335i $$0.119388\pi$$
−0.930483 + 0.366335i $$0.880612\pi$$
$$158$$ 1.90983i 0.151938i
$$159$$ −5.47214 −0.433969
$$160$$ 0 0
$$161$$ −13.3262 −1.05025
$$162$$ − 0.618034i − 0.0485573i
$$163$$ 11.0000i 0.861586i 0.902451 + 0.430793i $$0.141766\pi$$
−0.902451 + 0.430793i $$0.858234\pi$$
$$164$$ −8.47214 −0.661563
$$165$$ 0 0
$$166$$ −1.09017 −0.0846136
$$167$$ 5.56231i 0.430424i 0.976567 + 0.215212i $$0.0690443\pi$$
−0.976567 + 0.215212i $$0.930956\pi$$
$$168$$ 3.61803i 0.279137i
$$169$$ −10.5623 −0.812485
$$170$$ 0 0
$$171$$ 11.7082 0.895349
$$172$$ 3.00000i 0.228748i
$$173$$ − 16.8885i − 1.28401i −0.766700 0.642006i $$-0.778102\pi$$
0.766700 0.642006i $$-0.221898\pi$$
$$174$$ 0.854102 0.0647493
$$175$$ 0 0
$$176$$ −1.41641 −0.106766
$$177$$ 4.14590i 0.311625i
$$178$$ 5.52786i 0.414331i
$$179$$ 9.47214 0.707981 0.353990 0.935249i $$-0.384825\pi$$
0.353990 + 0.935249i $$0.384825\pi$$
$$180$$ 0 0
$$181$$ 13.7082 1.01892 0.509461 0.860494i $$-0.329845\pi$$
0.509461 + 0.860494i $$0.329845\pi$$
$$182$$ − 4.85410i − 0.359810i
$$183$$ − 4.70820i − 0.348040i
$$184$$ 18.4164 1.35768
$$185$$ 0 0
$$186$$ −1.85410 −0.135949
$$187$$ − 0.583592i − 0.0426765i
$$188$$ 2.61803i 0.190940i
$$189$$ −8.09017 −0.588473
$$190$$ 0 0
$$191$$ −24.1803 −1.74963 −0.874814 0.484459i $$-0.839016\pi$$
−0.874814 + 0.484459i $$0.839016\pi$$
$$192$$ 0.236068i 0.0170367i
$$193$$ − 5.70820i − 0.410886i −0.978669 0.205443i $$-0.934137\pi$$
0.978669 0.205443i $$-0.0658634\pi$$
$$194$$ −1.76393 −0.126643
$$195$$ 0 0
$$196$$ 7.09017 0.506441
$$197$$ 9.70820i 0.691681i 0.938293 + 0.345840i $$0.112406\pi$$
−0.938293 + 0.345840i $$0.887594\pi$$
$$198$$ 0.944272i 0.0671065i
$$199$$ −2.56231 −0.181637 −0.0908185 0.995867i $$-0.528948\pi$$
−0.0908185 + 0.995867i $$0.528948\pi$$
$$200$$ 0 0
$$201$$ 9.23607 0.651462
$$202$$ 4.61803i 0.324924i
$$203$$ 2.23607i 0.156941i
$$204$$ −1.23607 −0.0865421
$$205$$ 0 0
$$206$$ −7.14590 −0.497878
$$207$$ 16.4721i 1.14489i
$$208$$ − 9.00000i − 0.624038i
$$209$$ −4.47214 −0.309344
$$210$$ 0 0
$$211$$ 13.1803 0.907372 0.453686 0.891162i $$-0.350109\pi$$
0.453686 + 0.891162i $$0.350109\pi$$
$$212$$ 8.85410i 0.608102i
$$213$$ − 4.38197i − 0.300247i
$$214$$ −6.43769 −0.440072
$$215$$ 0 0
$$216$$ 11.1803 0.760726
$$217$$ − 4.85410i − 0.329518i
$$218$$ 6.18034i 0.418585i
$$219$$ 9.00000 0.608164
$$220$$ 0 0
$$221$$ 3.70820 0.249441
$$222$$ − 2.61803i − 0.175711i
$$223$$ 22.1803i 1.48531i 0.669677 + 0.742653i $$0.266433\pi$$
−0.669677 + 0.742653i $$0.733567\pi$$
$$224$$ 9.09017 0.607363
$$225$$ 0 0
$$226$$ 6.27051 0.417108
$$227$$ − 19.2361i − 1.27674i −0.769729 0.638371i $$-0.779608\pi$$
0.769729 0.638371i $$-0.220392\pi$$
$$228$$ 9.47214i 0.627308i
$$229$$ −8.29180 −0.547937 −0.273969 0.961739i $$-0.588336\pi$$
−0.273969 + 0.961739i $$0.588336\pi$$
$$230$$ 0 0
$$231$$ 1.23607 0.0813273
$$232$$ − 3.09017i − 0.202880i
$$233$$ 14.9443i 0.979032i 0.871994 + 0.489516i $$0.162826\pi$$
−0.871994 + 0.489516i $$0.837174\pi$$
$$234$$ −6.00000 −0.392232
$$235$$ 0 0
$$236$$ 6.70820 0.436667
$$237$$ − 3.09017i − 0.200728i
$$238$$ 0.763932i 0.0495184i
$$239$$ 29.4721 1.90639 0.953197 0.302350i $$-0.0977711\pi$$
0.953197 + 0.302350i $$0.0977711\pi$$
$$240$$ 0 0
$$241$$ 11.4721 0.738985 0.369493 0.929234i $$-0.379532\pi$$
0.369493 + 0.929234i $$0.379532\pi$$
$$242$$ 6.43769i 0.413831i
$$243$$ 16.0000i 1.02640i
$$244$$ −7.61803 −0.487695
$$245$$ 0 0
$$246$$ −3.23607 −0.206324
$$247$$ − 28.4164i − 1.80809i
$$248$$ 6.70820i 0.425971i
$$249$$ 1.76393 0.111785
$$250$$ 0 0
$$251$$ −6.81966 −0.430453 −0.215227 0.976564i $$-0.569049\pi$$
−0.215227 + 0.976564i $$0.569049\pi$$
$$252$$ 5.23607i 0.329841i
$$253$$ − 6.29180i − 0.395562i
$$254$$ 9.81966 0.616140
$$255$$ 0 0
$$256$$ −6.56231 −0.410144
$$257$$ − 16.1459i − 1.00715i −0.863951 0.503577i $$-0.832017\pi$$
0.863951 0.503577i $$-0.167983\pi$$
$$258$$ 1.14590i 0.0713405i
$$259$$ 6.85410 0.425893
$$260$$ 0 0
$$261$$ 2.76393 0.171083
$$262$$ 11.0000i 0.679582i
$$263$$ − 22.0902i − 1.36214i −0.732219 0.681069i $$-0.761515\pi$$
0.732219 0.681069i $$-0.238485\pi$$
$$264$$ −1.70820 −0.105133
$$265$$ 0 0
$$266$$ 5.85410 0.358938
$$267$$ − 8.94427i − 0.547381i
$$268$$ − 14.9443i − 0.912867i
$$269$$ 17.2361 1.05090 0.525451 0.850824i $$-0.323897\pi$$
0.525451 + 0.850824i $$0.323897\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ 1.41641i 0.0858823i
$$273$$ 7.85410i 0.475352i
$$274$$ −3.67376 −0.221940
$$275$$ 0 0
$$276$$ −13.3262 −0.802145
$$277$$ 11.2918i 0.678458i 0.940704 + 0.339229i $$0.110166\pi$$
−0.940704 + 0.339229i $$0.889834\pi$$
$$278$$ 3.09017i 0.185336i
$$279$$ −6.00000 −0.359211
$$280$$ 0 0
$$281$$ −1.09017 −0.0650341 −0.0325170 0.999471i $$-0.510352\pi$$
−0.0325170 + 0.999471i $$0.510352\pi$$
$$282$$ 1.00000i 0.0595491i
$$283$$ − 23.1459i − 1.37588i −0.725767 0.687940i $$-0.758515\pi$$
0.725767 0.687940i $$-0.241485\pi$$
$$284$$ −7.09017 −0.420724
$$285$$ 0 0
$$286$$ 2.29180 0.135517
$$287$$ − 8.47214i − 0.500094i
$$288$$ − 11.2361i − 0.662092i
$$289$$ 16.4164 0.965671
$$290$$ 0 0
$$291$$ 2.85410 0.167310
$$292$$ − 14.5623i − 0.852194i
$$293$$ − 28.4721i − 1.66336i −0.555255 0.831680i $$-0.687379\pi$$
0.555255 0.831680i $$-0.312621\pi$$
$$294$$ 2.70820 0.157946
$$295$$ 0 0
$$296$$ −9.47214 −0.550557
$$297$$ − 3.81966i − 0.221639i
$$298$$ 8.61803i 0.499229i
$$299$$ 39.9787 2.31203
$$300$$ 0 0
$$301$$ −3.00000 −0.172917
$$302$$ 3.43769i 0.197817i
$$303$$ − 7.47214i − 0.429263i
$$304$$ 10.8541 0.622525
$$305$$ 0 0
$$306$$ 0.944272 0.0539804
$$307$$ − 4.76393i − 0.271892i −0.990716 0.135946i $$-0.956593\pi$$
0.990716 0.135946i $$-0.0434074\pi$$
$$308$$ − 2.00000i − 0.113961i
$$309$$ 11.5623 0.657757
$$310$$ 0 0
$$311$$ −29.5066 −1.67316 −0.836582 0.547841i $$-0.815450\pi$$
−0.836582 + 0.547841i $$0.815450\pi$$
$$312$$ − 10.8541i − 0.614493i
$$313$$ − 21.2361i − 1.20033i −0.799875 0.600167i $$-0.795101\pi$$
0.799875 0.600167i $$-0.204899\pi$$
$$314$$ 5.67376 0.320189
$$315$$ 0 0
$$316$$ −5.00000 −0.281272
$$317$$ 23.6525i 1.32846i 0.747530 + 0.664228i $$0.231239\pi$$
−0.747530 + 0.664228i $$0.768761\pi$$
$$318$$ 3.38197i 0.189651i
$$319$$ −1.05573 −0.0591094
$$320$$ 0 0
$$321$$ 10.4164 0.581387
$$322$$ 8.23607i 0.458978i
$$323$$ 4.47214i 0.248836i
$$324$$ 1.61803 0.0898908
$$325$$ 0 0
$$326$$ 6.79837 0.376527
$$327$$ − 10.0000i − 0.553001i
$$328$$ 11.7082i 0.646477i
$$329$$ −2.61803 −0.144337
$$330$$ 0 0
$$331$$ 17.1246 0.941254 0.470627 0.882332i $$-0.344028\pi$$
0.470627 + 0.882332i $$0.344028\pi$$
$$332$$ − 2.85410i − 0.156639i
$$333$$ − 8.47214i − 0.464270i
$$334$$ 3.43769 0.188102
$$335$$ 0 0
$$336$$ −3.00000 −0.163663
$$337$$ − 1.14590i − 0.0624210i −0.999513 0.0312105i $$-0.990064\pi$$
0.999513 0.0312105i $$-0.00993623\pi$$
$$338$$ 6.52786i 0.355069i
$$339$$ −10.1459 −0.551050
$$340$$ 0 0
$$341$$ 2.29180 0.124108
$$342$$ − 7.23607i − 0.391282i
$$343$$ 18.4164i 0.994393i
$$344$$ 4.14590 0.223532
$$345$$ 0 0
$$346$$ −10.4377 −0.561134
$$347$$ 31.0902i 1.66901i 0.551002 + 0.834504i $$0.314246\pi$$
−0.551002 + 0.834504i $$0.685754\pi$$
$$348$$ 2.23607i 0.119866i
$$349$$ −8.29180 −0.443850 −0.221925 0.975064i $$-0.571234\pi$$
−0.221925 + 0.975064i $$0.571234\pi$$
$$350$$ 0 0
$$351$$ 24.2705 1.29546
$$352$$ 4.29180i 0.228753i
$$353$$ 24.0902i 1.28219i 0.767461 + 0.641095i $$0.221520\pi$$
−0.767461 + 0.641095i $$0.778480\pi$$
$$354$$ 2.56231 0.136185
$$355$$ 0 0
$$356$$ −14.4721 −0.767022
$$357$$ − 1.23607i − 0.0654197i
$$358$$ − 5.85410i − 0.309399i
$$359$$ 28.7426 1.51698 0.758489 0.651685i $$-0.225938\pi$$
0.758489 + 0.651685i $$0.225938\pi$$
$$360$$ 0 0
$$361$$ 15.2705 0.803711
$$362$$ − 8.47214i − 0.445286i
$$363$$ − 10.4164i − 0.546720i
$$364$$ 12.7082 0.666091
$$365$$ 0 0
$$366$$ −2.90983 −0.152099
$$367$$ 5.43769i 0.283845i 0.989878 + 0.141923i $$0.0453284\pi$$
−0.989878 + 0.141923i $$0.954672\pi$$
$$368$$ 15.2705i 0.796030i
$$369$$ −10.4721 −0.545158
$$370$$ 0 0
$$371$$ −8.85410 −0.459682
$$372$$ − 4.85410i − 0.251673i
$$373$$ 5.27051i 0.272897i 0.990647 + 0.136448i $$0.0435688\pi$$
−0.990647 + 0.136448i $$0.956431\pi$$
$$374$$ −0.360680 −0.0186503
$$375$$ 0 0
$$376$$ 3.61803 0.186586
$$377$$ − 6.70820i − 0.345490i
$$378$$ 5.00000i 0.257172i
$$379$$ 34.5967 1.77712 0.888558 0.458765i $$-0.151708\pi$$
0.888558 + 0.458765i $$0.151708\pi$$
$$380$$ 0 0
$$381$$ −15.8885 −0.813995
$$382$$ 14.9443i 0.764615i
$$383$$ − 11.3607i − 0.580504i −0.956950 0.290252i $$-0.906261\pi$$
0.956950 0.290252i $$-0.0937391\pi$$
$$384$$ 11.3820 0.580834
$$385$$ 0 0
$$386$$ −3.52786 −0.179564
$$387$$ 3.70820i 0.188499i
$$388$$ − 4.61803i − 0.234445i
$$389$$ −15.0000 −0.760530 −0.380265 0.924878i $$-0.624167\pi$$
−0.380265 + 0.924878i $$0.624167\pi$$
$$390$$ 0 0
$$391$$ −6.29180 −0.318190
$$392$$ − 9.79837i − 0.494893i
$$393$$ − 17.7984i − 0.897809i
$$394$$ 6.00000 0.302276
$$395$$ 0 0
$$396$$ −2.47214 −0.124230
$$397$$ 0.0344419i 0.00172859i 1.00000 0.000864294i $$0.000275113\pi$$
−1.00000 0.000864294i $$0.999725\pi$$
$$398$$ 1.58359i 0.0793783i
$$399$$ −9.47214 −0.474200
$$400$$ 0 0
$$401$$ −22.5967 −1.12843 −0.564214 0.825629i $$-0.690821\pi$$
−0.564214 + 0.825629i $$0.690821\pi$$
$$402$$ − 5.70820i − 0.284699i
$$403$$ 14.5623i 0.725400i
$$404$$ −12.0902 −0.601508
$$405$$ 0 0
$$406$$ 1.38197 0.0685858
$$407$$ 3.23607i 0.160406i
$$408$$ 1.70820i 0.0845687i
$$409$$ −28.4164 −1.40510 −0.702550 0.711634i $$-0.747955\pi$$
−0.702550 + 0.711634i $$0.747955\pi$$
$$410$$ 0 0
$$411$$ 5.94427 0.293209
$$412$$ − 18.7082i − 0.921687i
$$413$$ 6.70820i 0.330089i
$$414$$ 10.1803 0.500336
$$415$$ 0 0
$$416$$ −27.2705 −1.33705
$$417$$ − 5.00000i − 0.244851i
$$418$$ 2.76393i 0.135188i
$$419$$ 0.527864 0.0257878 0.0128939 0.999917i $$-0.495896\pi$$
0.0128939 + 0.999917i $$0.495896\pi$$
$$420$$ 0 0
$$421$$ 32.0000 1.55958 0.779792 0.626038i $$-0.215325\pi$$
0.779792 + 0.626038i $$0.215325\pi$$
$$422$$ − 8.14590i − 0.396536i
$$423$$ 3.23607i 0.157343i
$$424$$ 12.2361 0.594236
$$425$$ 0 0
$$426$$ −2.70820 −0.131213
$$427$$ − 7.61803i − 0.368663i
$$428$$ − 16.8541i − 0.814674i
$$429$$ −3.70820 −0.179034
$$430$$ 0 0
$$431$$ 23.8328 1.14799 0.573993 0.818860i $$-0.305394\pi$$
0.573993 + 0.818860i $$0.305394\pi$$
$$432$$ 9.27051i 0.446028i
$$433$$ 20.1459i 0.968150i 0.875026 + 0.484075i $$0.160844\pi$$
−0.875026 + 0.484075i $$0.839156\pi$$
$$434$$ −3.00000 −0.144005
$$435$$ 0 0
$$436$$ −16.1803 −0.774898
$$437$$ 48.2148i 2.30643i
$$438$$ − 5.56231i − 0.265777i
$$439$$ −5.97871 −0.285348 −0.142674 0.989770i $$-0.545570\pi$$
−0.142674 + 0.989770i $$0.545570\pi$$
$$440$$ 0 0
$$441$$ 8.76393 0.417330
$$442$$ − 2.29180i − 0.109010i
$$443$$ 12.0557i 0.572785i 0.958112 + 0.286392i $$0.0924561\pi$$
−0.958112 + 0.286392i $$0.907544\pi$$
$$444$$ 6.85410 0.325281
$$445$$ 0 0
$$446$$ 13.7082 0.649102
$$447$$ − 13.9443i − 0.659541i
$$448$$ 0.381966i 0.0180462i
$$449$$ −20.3262 −0.959254 −0.479627 0.877472i $$-0.659228\pi$$
−0.479627 + 0.877472i $$0.659228\pi$$
$$450$$ 0 0
$$451$$ 4.00000 0.188353
$$452$$ 16.4164i 0.772163i
$$453$$ − 5.56231i − 0.261340i
$$454$$ −11.8885 −0.557957
$$455$$ 0 0
$$456$$ 13.0902 0.613003
$$457$$ − 5.41641i − 0.253369i −0.991943 0.126684i $$-0.959566\pi$$
0.991943 0.126684i $$-0.0404336\pi$$
$$458$$ 5.12461i 0.239457i
$$459$$ −3.81966 −0.178286
$$460$$ 0 0
$$461$$ 23.1803 1.07962 0.539808 0.841788i $$-0.318497\pi$$
0.539808 + 0.841788i $$0.318497\pi$$
$$462$$ − 0.763932i − 0.0355413i
$$463$$ 16.1246i 0.749374i 0.927151 + 0.374687i $$0.122250\pi$$
−0.927151 + 0.374687i $$0.877750\pi$$
$$464$$ 2.56231 0.118952
$$465$$ 0 0
$$466$$ 9.23607 0.427853
$$467$$ 28.4508i 1.31655i 0.752778 + 0.658274i $$0.228713\pi$$
−0.752778 + 0.658274i $$0.771287\pi$$
$$468$$ − 15.7082i − 0.726112i
$$469$$ 14.9443 0.690062
$$470$$ 0 0
$$471$$ −9.18034 −0.423008
$$472$$ − 9.27051i − 0.426710i
$$473$$ − 1.41641i − 0.0651265i
$$474$$ −1.90983 −0.0877214
$$475$$ 0 0
$$476$$ −2.00000 −0.0916698
$$477$$ 10.9443i 0.501104i
$$478$$ − 18.2148i − 0.833125i
$$479$$ 4.14590 0.189431 0.0947155 0.995504i $$-0.469806\pi$$
0.0947155 + 0.995504i $$0.469806\pi$$
$$480$$ 0 0
$$481$$ −20.5623 −0.937560
$$482$$ − 7.09017i − 0.322948i
$$483$$ − 13.3262i − 0.606365i
$$484$$ −16.8541 −0.766096
$$485$$ 0 0
$$486$$ 9.88854 0.448553
$$487$$ 9.58359i 0.434274i 0.976141 + 0.217137i $$0.0696718\pi$$
−0.976141 + 0.217137i $$0.930328\pi$$
$$488$$ 10.5279i 0.476574i
$$489$$ −11.0000 −0.497437
$$490$$ 0 0
$$491$$ 37.2492 1.68103 0.840517 0.541785i $$-0.182251\pi$$
0.840517 + 0.541785i $$0.182251\pi$$
$$492$$ − 8.47214i − 0.381953i
$$493$$ 1.05573i 0.0475476i
$$494$$ −17.5623 −0.790165
$$495$$ 0 0
$$496$$ −5.56231 −0.249755
$$497$$ − 7.09017i − 0.318038i
$$498$$ − 1.09017i − 0.0488517i
$$499$$ 12.5623 0.562366 0.281183 0.959654i $$-0.409273\pi$$
0.281183 + 0.959654i $$0.409273\pi$$
$$500$$ 0 0
$$501$$ −5.56231 −0.248506
$$502$$ 4.21478i 0.188115i
$$503$$ − 10.5836i − 0.471899i −0.971765 0.235950i $$-0.924180\pi$$
0.971765 0.235950i $$-0.0758200\pi$$
$$504$$ 7.23607 0.322320
$$505$$ 0 0
$$506$$ −3.88854 −0.172867
$$507$$ − 10.5623i − 0.469088i
$$508$$ 25.7082i 1.14062i
$$509$$ 4.67376 0.207161 0.103580 0.994621i $$-0.466970\pi$$
0.103580 + 0.994621i $$0.466970\pi$$
$$510$$ 0 0
$$511$$ 14.5623 0.644198
$$512$$ − 18.7082i − 0.826794i
$$513$$ 29.2705i 1.29232i
$$514$$ −9.97871 −0.440142
$$515$$ 0 0
$$516$$ −3.00000 −0.132068
$$517$$ − 1.23607i − 0.0543622i
$$518$$ − 4.23607i − 0.186122i
$$519$$ 16.8885 0.741325
$$520$$ 0 0
$$521$$ −15.3607 −0.672964 −0.336482 0.941690i $$-0.609237\pi$$
−0.336482 + 0.941690i $$0.609237\pi$$
$$522$$ − 1.70820i − 0.0747661i
$$523$$ − 19.8541i − 0.868159i −0.900875 0.434080i $$-0.857074\pi$$
0.900875 0.434080i $$-0.142926\pi$$
$$524$$ −28.7984 −1.25806
$$525$$ 0 0
$$526$$ −13.6525 −0.595276
$$527$$ − 2.29180i − 0.0998322i
$$528$$ − 1.41641i − 0.0616412i
$$529$$ −44.8328 −1.94925
$$530$$ 0 0
$$531$$ 8.29180 0.359833
$$532$$ 15.3262i 0.664477i
$$533$$ 25.4164i 1.10091i
$$534$$ −5.52786 −0.239214
$$535$$ 0 0
$$536$$ −20.6525 −0.892051
$$537$$ 9.47214i 0.408753i
$$538$$ − 10.6525i − 0.459261i
$$539$$ −3.34752 −0.144188
$$540$$ 0 0
$$541$$ −13.1246 −0.564271 −0.282136 0.959375i $$-0.591043\pi$$
−0.282136 + 0.959375i $$0.591043\pi$$
$$542$$ 4.94427i 0.212375i
$$543$$ 13.7082i 0.588275i
$$544$$ 4.29180 0.184009
$$545$$ 0 0
$$546$$ 4.85410 0.207736
$$547$$ 34.7082i 1.48402i 0.670391 + 0.742008i $$0.266126\pi$$
−0.670391 + 0.742008i $$0.733874\pi$$
$$548$$ − 9.61803i − 0.410862i
$$549$$ −9.41641 −0.401882
$$550$$ 0 0
$$551$$ 8.09017 0.344653
$$552$$ 18.4164i 0.783854i
$$553$$ − 5.00000i − 0.212622i
$$554$$ 6.97871 0.296497
$$555$$ 0 0
$$556$$ −8.09017 −0.343100
$$557$$ − 9.23607i − 0.391345i −0.980669 0.195672i $$-0.937311\pi$$
0.980669 0.195672i $$-0.0626889\pi$$
$$558$$ 3.70820i 0.156981i
$$559$$ 9.00000 0.380659
$$560$$ 0 0
$$561$$ 0.583592 0.0246393
$$562$$ 0.673762i 0.0284209i
$$563$$ 9.61803i 0.405352i 0.979246 + 0.202676i $$0.0649638\pi$$
−0.979246 + 0.202676i $$0.935036\pi$$
$$564$$ −2.61803 −0.110239
$$565$$ 0 0
$$566$$ −14.3050 −0.601282
$$567$$ 1.61803i 0.0679510i
$$568$$ 9.79837i 0.411131i
$$569$$ −29.4721 −1.23554 −0.617768 0.786360i $$-0.711963\pi$$
−0.617768 + 0.786360i $$0.711963\pi$$
$$570$$ 0 0
$$571$$ 32.1246 1.34437 0.672187 0.740382i $$-0.265355\pi$$
0.672187 + 0.740382i $$0.265355\pi$$
$$572$$ 6.00000i 0.250873i
$$573$$ − 24.1803i − 1.01015i
$$574$$ −5.23607 −0.218549
$$575$$ 0 0
$$576$$ 0.472136 0.0196723
$$577$$ − 37.7771i − 1.57268i −0.617794 0.786340i $$-0.711973\pi$$
0.617794 0.786340i $$-0.288027\pi$$
$$578$$ − 10.1459i − 0.422014i
$$579$$ 5.70820 0.237225
$$580$$ 0 0
$$581$$ 2.85410 0.118408
$$582$$ − 1.76393i − 0.0731173i
$$583$$ − 4.18034i − 0.173132i
$$584$$ −20.1246 −0.832762
$$585$$ 0 0
$$586$$ −17.5967 −0.726915
$$587$$ − 18.7082i − 0.772170i −0.922463 0.386085i $$-0.873827\pi$$
0.922463 0.386085i $$-0.126173\pi$$
$$588$$ 7.09017i 0.292394i
$$589$$ −17.5623 −0.723642
$$590$$ 0 0
$$591$$ −9.70820 −0.399342
$$592$$ − 7.85410i − 0.322802i
$$593$$ − 22.0902i − 0.907135i −0.891222 0.453567i $$-0.850151\pi$$
0.891222 0.453567i $$-0.149849\pi$$
$$594$$ −2.36068 −0.0968599
$$595$$ 0 0
$$596$$ −22.5623 −0.924188
$$597$$ − 2.56231i − 0.104868i
$$598$$ − 24.7082i − 1.01039i
$$599$$ 0.527864 0.0215679 0.0107840 0.999942i $$-0.496567\pi$$
0.0107840 + 0.999942i $$0.496567\pi$$
$$600$$ 0 0
$$601$$ 36.2705 1.47950 0.739752 0.672879i $$-0.234943\pi$$
0.739752 + 0.672879i $$0.234943\pi$$
$$602$$ 1.85410i 0.0755676i
$$603$$ − 18.4721i − 0.752244i
$$604$$ −9.00000 −0.366205
$$605$$ 0 0
$$606$$ −4.61803 −0.187595
$$607$$ 15.4377i 0.626597i 0.949655 + 0.313298i $$0.101434\pi$$
−0.949655 + 0.313298i $$0.898566\pi$$
$$608$$ − 32.8885i − 1.33381i
$$609$$ −2.23607 −0.0906100
$$610$$ 0 0
$$611$$ 7.85410 0.317743
$$612$$ 2.47214i 0.0999302i
$$613$$ 31.9787i 1.29161i 0.763503 + 0.645804i $$0.223478\pi$$
−0.763503 + 0.645804i $$0.776522\pi$$
$$614$$ −2.94427 −0.118821
$$615$$ 0 0
$$616$$ −2.76393 −0.111362
$$617$$ − 9.76393i − 0.393081i −0.980496 0.196541i $$-0.937029\pi$$
0.980496 0.196541i $$-0.0629707\pi$$
$$618$$ − 7.14590i − 0.287450i
$$619$$ −39.4721 −1.58652 −0.793260 0.608884i $$-0.791618\pi$$
−0.793260 + 0.608884i $$0.791618\pi$$
$$620$$ 0 0
$$621$$ −41.1803 −1.65251
$$622$$ 18.2361i 0.731200i
$$623$$ − 14.4721i − 0.579814i
$$624$$ 9.00000 0.360288
$$625$$ 0 0
$$626$$ −13.1246 −0.524565
$$627$$ − 4.47214i − 0.178600i
$$628$$ 14.8541i 0.592743i
$$629$$ 3.23607 0.129030
$$630$$ 0 0
$$631$$ −5.76393 −0.229459 −0.114729 0.993397i $$-0.536600\pi$$
−0.114729 + 0.993397i $$0.536600\pi$$
$$632$$ 6.90983i 0.274858i
$$633$$ 13.1803i 0.523871i
$$634$$ 14.6180 0.580556
$$635$$ 0 0
$$636$$ −8.85410 −0.351088
$$637$$ − 21.2705i − 0.842768i
$$638$$ 0.652476i 0.0258318i
$$639$$ −8.76393 −0.346696
$$640$$ 0 0
$$641$$ 10.0902 0.398538 0.199269 0.979945i $$-0.436143\pi$$
0.199269 + 0.979945i $$0.436143\pi$$
$$642$$ − 6.43769i − 0.254076i
$$643$$ 22.8328i 0.900438i 0.892918 + 0.450219i $$0.148654\pi$$
−0.892918 + 0.450219i $$0.851346\pi$$
$$644$$ −21.5623 −0.849674
$$645$$ 0 0
$$646$$ 2.76393 0.108745
$$647$$ − 30.5410i − 1.20069i −0.799741 0.600346i $$-0.795030\pi$$
0.799741 0.600346i $$-0.204970\pi$$
$$648$$ − 2.23607i − 0.0878410i
$$649$$ −3.16718 −0.124323
$$650$$ 0 0
$$651$$ 4.85410 0.190247
$$652$$ 17.7984i 0.697038i
$$653$$ 7.90983i 0.309536i 0.987951 + 0.154768i $$0.0494629\pi$$
−0.987951 + 0.154768i $$0.950537\pi$$
$$654$$ −6.18034 −0.241670
$$655$$ 0 0
$$656$$ −9.70820 −0.379042
$$657$$ − 18.0000i − 0.702247i
$$658$$ 1.61803i 0.0630775i
$$659$$ −24.4721 −0.953299 −0.476650 0.879093i $$-0.658149\pi$$
−0.476650 + 0.879093i $$0.658149\pi$$
$$660$$ 0 0
$$661$$ −40.6869 −1.58254 −0.791269 0.611468i $$-0.790579\pi$$
−0.791269 + 0.611468i $$0.790579\pi$$
$$662$$ − 10.5836i − 0.411343i
$$663$$ 3.70820i 0.144015i
$$664$$ −3.94427 −0.153067
$$665$$ 0 0
$$666$$ −5.23607 −0.202894
$$667$$ 11.3820i 0.440711i
$$668$$ 9.00000i 0.348220i
$$669$$ −22.1803 −0.857541
$$670$$ 0 0
$$671$$ 3.59675 0.138851
$$672$$ 9.09017i 0.350661i
$$673$$ − 10.1803i − 0.392423i −0.980562 0.196212i $$-0.937136\pi$$
0.980562 0.196212i $$-0.0628640\pi$$
$$674$$ −0.708204 −0.0272790
$$675$$ 0 0
$$676$$ −17.0902 −0.657314
$$677$$ − 8.38197i − 0.322145i −0.986943 0.161073i $$-0.948505\pi$$
0.986943 0.161073i $$-0.0514953\pi$$
$$678$$ 6.27051i 0.240817i
$$679$$ 4.61803 0.177224
$$680$$ 0 0
$$681$$ 19.2361 0.737128
$$682$$ − 1.41641i − 0.0542371i
$$683$$ − 4.52786i − 0.173254i −0.996241 0.0866270i $$-0.972391\pi$$
0.996241 0.0866270i $$-0.0276088\pi$$
$$684$$ 18.9443 0.724352
$$685$$ 0 0
$$686$$ 11.3820 0.434565
$$687$$ − 8.29180i − 0.316352i
$$688$$ 3.43769i 0.131061i
$$689$$ 26.5623 1.01194
$$690$$ 0 0
$$691$$ 2.72949 0.103835 0.0519173 0.998651i $$-0.483467\pi$$
0.0519173 + 0.998651i $$0.483467\pi$$
$$692$$ − 27.3262i − 1.03879i
$$693$$ − 2.47214i − 0.0939087i
$$694$$ 19.2148 0.729383
$$695$$ 0 0
$$696$$ 3.09017 0.117133
$$697$$ − 4.00000i − 0.151511i
$$698$$ 5.12461i 0.193969i
$$699$$ −14.9443 −0.565244
$$700$$ 0 0
$$701$$ 35.0132 1.32243 0.661214 0.750197i $$-0.270041\pi$$
0.661214 + 0.750197i $$0.270041\pi$$
$$702$$ − 15.0000i − 0.566139i
$$703$$ − 24.7984i − 0.935288i
$$704$$ −0.180340 −0.00679682
$$705$$ 0 0
$$706$$ 14.8885 0.560338
$$707$$ − 12.0902i − 0.454698i
$$708$$ 6.70820i 0.252110i
$$709$$ −33.5410 −1.25966 −0.629830 0.776733i $$-0.716875\pi$$
−0.629830 + 0.776733i $$0.716875\pi$$
$$710$$ 0 0
$$711$$ −6.18034 −0.231781
$$712$$ 20.0000i 0.749532i
$$713$$ − 24.7082i − 0.925330i
$$714$$ −0.763932 −0.0285894
$$715$$ 0 0
$$716$$ 15.3262 0.572768
$$717$$ 29.4721i 1.10066i
$$718$$ − 17.7639i − 0.662944i
$$719$$ 36.7082 1.36899 0.684493 0.729020i $$-0.260024\pi$$
0.684493 + 0.729020i $$0.260024\pi$$
$$720$$ 0 0
$$721$$ 18.7082 0.696730
$$722$$ − 9.43769i − 0.351235i
$$723$$ 11.4721i 0.426653i
$$724$$ 22.1803 0.824326
$$725$$ 0 0
$$726$$ −6.43769 −0.238925
$$727$$ − 4.43769i − 0.164585i −0.996608 0.0822925i $$-0.973776\pi$$
0.996608 0.0822925i $$-0.0262242\pi$$
$$728$$ − 17.5623i − 0.650902i
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ −1.41641 −0.0523877
$$732$$ − 7.61803i − 0.281571i
$$733$$ 26.9787i 0.996482i 0.867039 + 0.498241i $$0.166020\pi$$
−0.867039 + 0.498241i $$0.833980\pi$$
$$734$$ 3.36068 0.124045
$$735$$ 0 0
$$736$$ 46.2705 1.70555
$$737$$ 7.05573i 0.259901i
$$738$$ 6.47214i 0.238243i
$$739$$ 30.9787 1.13957 0.569785 0.821794i $$-0.307026\pi$$
0.569785 + 0.821794i $$0.307026\pi$$
$$740$$ 0 0
$$741$$ 28.4164 1.04390
$$742$$ 5.47214i 0.200888i
$$743$$ − 16.3607i − 0.600215i −0.953905 0.300108i $$-0.902977\pi$$
0.953905 0.300108i $$-0.0970226\pi$$
$$744$$ −6.70820 −0.245935
$$745$$ 0 0
$$746$$ 3.25735 0.119260
$$747$$ − 3.52786i − 0.129078i
$$748$$ − 0.944272i − 0.0345260i
$$749$$ 16.8541 0.615835
$$750$$ 0 0
$$751$$ −40.8885 −1.49204 −0.746022 0.665921i $$-0.768039\pi$$
−0.746022 + 0.665921i $$0.768039\pi$$
$$752$$ 3.00000i 0.109399i
$$753$$ − 6.81966i − 0.248522i
$$754$$ −4.14590 −0.150985
$$755$$ 0 0
$$756$$ −13.0902 −0.476085
$$757$$ − 3.58359i − 0.130248i −0.997877 0.0651239i $$-0.979256\pi$$
0.997877 0.0651239i $$-0.0207443\pi$$
$$758$$ − 21.3820i − 0.776628i
$$759$$ 6.29180 0.228378
$$760$$ 0 0
$$761$$ 37.4508 1.35759 0.678796 0.734327i $$-0.262502\pi$$
0.678796 + 0.734327i $$0.262502\pi$$
$$762$$ 9.81966i 0.355729i
$$763$$ − 16.1803i − 0.585768i
$$764$$ −39.1246 −1.41548
$$765$$ 0 0
$$766$$ −7.02129 −0.253689
$$767$$ − 20.1246i − 0.726658i
$$768$$ − 6.56231i − 0.236797i
$$769$$ 13.4164 0.483808 0.241904 0.970300i $$-0.422228\pi$$
0.241904 + 0.970300i $$0.422228\pi$$
$$770$$ 0 0
$$771$$ 16.1459 0.581480
$$772$$ − 9.23607i − 0.332413i
$$773$$ 33.1591i 1.19265i 0.802744 + 0.596324i $$0.203373\pi$$
−0.802744 + 0.596324i $$0.796627\pi$$
$$774$$ 2.29180 0.0823769
$$775$$ 0 0
$$776$$ −6.38197 −0.229099
$$777$$ 6.85410i 0.245890i
$$778$$ 9.27051i 0.332364i
$$779$$ −30.6525 −1.09824
$$780$$ 0 0
$$781$$ 3.34752 0.119784
$$782$$ 3.88854i 0.139054i
$$783$$ 6.90983i 0.246937i
$$784$$ 8.12461 0.290165
$$785$$ 0 0
$$786$$ −11.0000 −0.392357
$$787$$ 34.1803i 1.21840i 0.793018 + 0.609199i $$0.208509\pi$$
−0.793018 + 0.609199i $$0.791491\pi$$
$$788$$ 15.7082i 0.559582i
$$789$$ 22.0902 0.786431
$$790$$ 0 0
$$791$$ −16.4164 −0.583700
$$792$$ 3.41641i 0.121397i
$$793$$ 22.8541i 0.811573i
$$794$$ 0.0212862 0.000755420 0
$$795$$ 0 0
$$796$$ −4.14590 −0.146947
$$797$$ − 14.2361i − 0.504267i −0.967692 0.252134i $$-0.918868\pi$$
0.967692 0.252134i $$-0.0811323\pi$$
$$798$$ 5.85410i 0.207233i
$$799$$ −1.23607 −0.0437289
$$800$$ 0 0
$$801$$ −17.8885 −0.632061
$$802$$ 13.9656i 0.493141i
$$803$$ 6.87539i 0.242627i
$$804$$ 14.9443 0.527044
$$805$$ 0 0
$$806$$ 9.00000 0.317011
$$807$$ 17.2361i 0.606738i
$$808$$ 16.7082i 0.587793i
$$809$$ 15.9787 0.561782 0.280891 0.959740i $$-0.409370\pi$$
0.280891 + 0.959740i $$0.409370\pi$$
$$810$$ 0 0
$$811$$ −1.29180 −0.0453611 −0.0226805 0.999743i $$-0.507220\pi$$
−0.0226805 + 0.999743i $$0.507220\pi$$
$$812$$ 3.61803i 0.126968i
$$813$$ − 8.00000i − 0.280572i
$$814$$ 2.00000 0.0701000
$$815$$ 0 0
$$816$$ −1.41641 −0.0495842
$$817$$ 10.8541i 0.379737i
$$818$$ 17.5623i 0.614052i
$$819$$ 15.7082 0.548889
$$820$$ 0 0
$$821$$ 19.6869 0.687078 0.343539 0.939138i $$-0.388374\pi$$
0.343539 + 0.939138i $$0.388374\pi$$
$$822$$ − 3.67376i − 0.128137i
$$823$$ 34.2918i 1.19534i 0.801743 + 0.597668i $$0.203906\pi$$
−0.801743 + 0.597668i $$0.796094\pi$$
$$824$$ −25.8541 −0.900670
$$825$$ 0 0
$$826$$ 4.14590 0.144254
$$827$$ 30.0344i 1.04440i 0.852823 + 0.522200i $$0.174889\pi$$
−0.852823 + 0.522200i $$0.825111\pi$$
$$828$$ 26.6525i 0.926238i
$$829$$ 29.1459 1.01228 0.506139 0.862452i $$-0.331072\pi$$
0.506139 + 0.862452i $$0.331072\pi$$
$$830$$ 0 0
$$831$$ −11.2918 −0.391708
$$832$$ − 1.14590i − 0.0397269i
$$833$$ 3.34752i 0.115985i
$$834$$ −3.09017 −0.107004
$$835$$ 0 0
$$836$$ −7.23607 −0.250265
$$837$$ − 15.0000i − 0.518476i
$$838$$ − 0.326238i − 0.0112697i
$$839$$ −4.14590 −0.143132 −0.0715661 0.997436i $$-0.522800\pi$$
−0.0715661 + 0.997436i $$0.522800\pi$$
$$840$$ 0 0
$$841$$ −27.0902 −0.934144
$$842$$ − 19.7771i − 0.681563i
$$843$$ − 1.09017i − 0.0375474i
$$844$$ 21.3262 0.734079
$$845$$ 0 0
$$846$$ 2.00000 0.0687614
$$847$$ − 16.8541i − 0.579114i
$$848$$ 10.1459i 0.348412i
$$849$$ 23.1459 0.794365
$$850$$ 0 0
$$851$$ 34.8885 1.19596
$$852$$ − 7.09017i − 0.242905i
$$853$$ 47.3050i 1.61969i 0.586643 + 0.809845i $$0.300449\pi$$
−0.586643 + 0.809845i $$0.699551\pi$$
$$854$$ −4.70820 −0.161111
$$855$$ 0 0
$$856$$ −23.2918 −0.796097
$$857$$ 40.6869i 1.38984i 0.719088 + 0.694919i $$0.244560\pi$$
−0.719088 + 0.694919i $$0.755440\pi$$
$$858$$ 2.29180i 0.0782406i
$$859$$ 28.4164 0.969555 0.484778 0.874637i $$-0.338901\pi$$
0.484778 + 0.874637i $$0.338901\pi$$
$$860$$ 0 0
$$861$$ 8.47214 0.288730
$$862$$ − 14.7295i − 0.501688i
$$863$$ − 41.5623i − 1.41480i −0.706815 0.707399i $$-0.749869\pi$$
0.706815 0.707399i $$-0.250131\pi$$
$$864$$ 28.0902 0.955647
$$865$$ 0 0
$$866$$ 12.4508 0.423097
$$867$$ 16.4164i 0.557530i
$$868$$ − 7.85410i − 0.266586i
$$869$$ 2.36068 0.0800806
$$870$$ 0 0
$$871$$ −44.8328 −1.51910
$$872$$ 22.3607i 0.757228i
$$873$$ − 5.70820i − 0.193193i
$$874$$ 29.7984 1.00795
$$875$$ 0 0
$$876$$ 14.5623 0.492015
$$877$$ − 30.5410i − 1.03130i −0.856800 0.515648i $$-0.827551\pi$$
0.856800 0.515648i $$-0.172449\pi$$
$$878$$ 3.69505i 0.124702i
$$879$$ 28.4721 0.960341
$$880$$ 0 0
$$881$$ 4.36068 0.146915 0.0734575 0.997298i $$-0.476597\pi$$
0.0734575 + 0.997298i $$0.476597\pi$$
$$882$$ − 5.41641i − 0.182380i
$$883$$ − 47.4164i − 1.59569i −0.602863 0.797845i $$-0.705974\pi$$
0.602863 0.797845i $$-0.294026\pi$$
$$884$$ 6.00000 0.201802
$$885$$ 0 0
$$886$$ 7.45085 0.250316
$$887$$ 5.88854i 0.197718i 0.995101 + 0.0988590i $$0.0315193\pi$$
−0.995101 + 0.0988590i $$0.968481\pi$$
$$888$$ − 9.47214i − 0.317864i
$$889$$ −25.7082 −0.862225
$$890$$ 0 0
$$891$$ −0.763932 −0.0255927
$$892$$ 35.8885i 1.20164i
$$893$$ 9.47214i 0.316973i
$$894$$ −8.61803 −0.288230
$$895$$ 0 0
$$896$$ 18.4164 0.615249
$$897$$ 39.9787i 1.33485i
$$898$$ 12.5623i 0.419210i
$$899$$ −4.14590 −0.138273
$$900$$ 0 0
$$901$$ −4.18034 −0.139267
$$902$$ − 2.47214i − 0.0823131i
$$903$$ − 3.00000i − 0.0998337i
$$904$$ 22.6869 0.754556
$$905$$ 0 0
$$906$$ −3.43769 −0.114210
$$907$$ − 47.2492i − 1.56888i −0.620202 0.784442i $$-0.712949\pi$$
0.620202 0.784442i $$-0.287051\pi$$
$$908$$ − 31.1246i − 1.03291i
$$909$$ −14.9443 −0.495670
$$910$$ 0 0
$$911$$ −35.7639 −1.18491 −0.592456 0.805603i $$-0.701842\pi$$
−0.592456 + 0.805603i $$0.701842\pi$$
$$912$$ 10.8541i 0.359415i
$$913$$ 1.34752i 0.0445965i
$$914$$ −3.34752 −0.110726
$$915$$ 0 0
$$916$$ −13.4164 −0.443291
$$917$$ − 28.7984i − 0.951006i
$$918$$ 2.36068i 0.0779140i
$$919$$ 1.78522 0.0588889 0.0294445 0.999566i $$-0.490626\pi$$
0.0294445 + 0.999566i $$0.490626\pi$$
$$920$$ 0 0
$$921$$ 4.76393 0.156977
$$922$$ − 14.3262i − 0.471810i
$$923$$ 21.2705i 0.700127i
$$924$$ 2.00000 0.0657952
$$925$$ 0 0
$$926$$ 9.96556 0.327489
$$927$$ − 23.1246i − 0.759512i
$$928$$ − 7.76393i − 0.254864i
$$929$$ 36.6312 1.20183 0.600915 0.799313i $$-0.294803\pi$$
0.600915 + 0.799313i $$0.294803\pi$$
$$930$$ 0 0
$$931$$ 25.6525 0.840726
$$932$$ 24.1803i 0.792053i
$$933$$ − 29.5066i − 0.966002i
$$934$$ 17.5836 0.575353
$$935$$ 0 0
$$936$$ −21.7082 −0.709555
$$937$$ − 51.2705i − 1.67493i −0.546487 0.837467i $$-0.684035\pi$$
0.546487 0.837467i $$-0.315965\pi$$
$$938$$ − 9.23607i − 0.301568i
$$939$$ 21.2361 0.693013
$$940$$ 0 0
$$941$$ −19.5836 −0.638407 −0.319203 0.947686i $$-0.603415\pi$$
−0.319203 + 0.947686i $$0.603415\pi$$
$$942$$ 5.67376i 0.184861i
$$943$$ − 43.1246i − 1.40433i
$$944$$ 7.68692 0.250188
$$945$$ 0 0
$$946$$ −0.875388 −0.0284613
$$947$$ 28.6525i 0.931080i 0.885027 + 0.465540i $$0.154140\pi$$
−0.885027 + 0.465540i $$0.845860\pi$$
$$948$$ − 5.00000i − 0.162392i
$$949$$ −43.6869 −1.41814
$$950$$ 0 0
$$951$$ −23.6525 −0.766984
$$952$$ 2.76393i 0.0895796i
$$953$$ 34.7426i 1.12542i 0.826653 + 0.562712i $$0.190242\pi$$
−0.826653 + 0.562712i $$0.809758\pi$$
$$954$$ 6.76393 0.218990
$$955$$ 0 0
$$956$$ 47.6869 1.54231
$$957$$ − 1.05573i − 0.0341268i
$$958$$ − 2.56231i − 0.0827843i
$$959$$ 9.61803 0.310583
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ 12.7082i 0.409729i
$$963$$ − 20.8328i − 0.671328i
$$964$$ 18.5623 0.597852
$$965$$ 0 0
$$966$$ −8.23607 −0.264991
$$967$$ − 39.8885i − 1.28273i −0.767236 0.641365i $$-0.778369\pi$$
0.767236 0.641365i $$-0.221631\pi$$
$$968$$ 23.2918i 0.748627i
$$969$$ −4.47214 −0.143666
$$970$$ 0 0
$$971$$ 3.38197 0.108532 0.0542662 0.998527i $$-0.482718\pi$$
0.0542662 + 0.998527i $$0.482718\pi$$
$$972$$ 25.8885i 0.830375i
$$973$$ − 8.09017i − 0.259359i
$$974$$ 5.92299 0.189785
$$975$$ 0 0
$$976$$ −8.72949 −0.279424
$$977$$ 33.6525i 1.07664i 0.842741 + 0.538319i $$0.180940\pi$$
−0.842741 + 0.538319i $$0.819060\pi$$
$$978$$ 6.79837i 0.217388i
$$979$$ 6.83282 0.218378
$$980$$ 0 0
$$981$$ −20.0000 −0.638551
$$982$$ − 23.0213i − 0.734639i
$$983$$ 7.38197i 0.235448i 0.993046 + 0.117724i $$0.0375598\pi$$
−0.993046 + 0.117724i $$0.962440\pi$$
$$984$$ −11.7082 −0.373244
$$985$$ 0 0
$$986$$ 0.652476 0.0207791
$$987$$ − 2.61803i − 0.0833329i
$$988$$ − 45.9787i − 1.46278i
$$989$$ −15.2705 −0.485574
$$990$$ 0 0
$$991$$ 29.3607 0.932673 0.466336 0.884607i $$-0.345574\pi$$
0.466336 + 0.884607i $$0.345574\pi$$
$$992$$ 16.8541i 0.535118i
$$993$$ 17.1246i 0.543433i
$$994$$ −4.38197 −0.138988
$$995$$ 0 0
$$996$$ 2.85410 0.0904357
$$997$$ 10.8885i 0.344844i 0.985023 + 0.172422i $$0.0551592\pi$$
−0.985023 + 0.172422i $$0.944841\pi$$
$$998$$ − 7.76393i − 0.245763i
$$999$$ 21.1803 0.670116
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 625.2.b.a.624.2 4
5.2 odd 4 625.2.a.b.1.2 2
5.3 odd 4 625.2.a.c.1.1 2
5.4 even 2 inner 625.2.b.a.624.3 4
15.2 even 4 5625.2.a.f.1.1 2
15.8 even 4 5625.2.a.d.1.2 2
20.3 even 4 10000.2.a.l.1.1 2
20.7 even 4 10000.2.a.c.1.2 2
25.2 odd 20 625.2.d.h.501.1 4
25.3 odd 20 125.2.d.a.76.1 4
25.4 even 10 125.2.e.a.49.2 8
25.6 even 5 125.2.e.a.74.2 8
25.8 odd 20 125.2.d.a.51.1 4
25.9 even 10 625.2.e.c.499.1 8
25.11 even 5 625.2.e.c.124.1 8
25.12 odd 20 625.2.d.h.126.1 4
25.13 odd 20 625.2.d.b.126.1 4
25.14 even 10 625.2.e.c.124.2 8
25.16 even 5 625.2.e.c.499.2 8
25.17 odd 20 25.2.d.a.11.1 4
25.19 even 10 125.2.e.a.74.1 8
25.21 even 5 125.2.e.a.49.1 8
25.22 odd 20 25.2.d.a.16.1 yes 4
25.23 odd 20 625.2.d.b.501.1 4
75.17 even 20 225.2.h.b.136.1 4
75.47 even 20 225.2.h.b.91.1 4
100.47 even 20 400.2.u.b.241.1 4
100.67 even 20 400.2.u.b.161.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
25.2.d.a.11.1 4 25.17 odd 20
25.2.d.a.16.1 yes 4 25.22 odd 20
125.2.d.a.51.1 4 25.8 odd 20
125.2.d.a.76.1 4 25.3 odd 20
125.2.e.a.49.1 8 25.21 even 5
125.2.e.a.49.2 8 25.4 even 10
125.2.e.a.74.1 8 25.19 even 10
125.2.e.a.74.2 8 25.6 even 5
225.2.h.b.91.1 4 75.47 even 20
225.2.h.b.136.1 4 75.17 even 20
400.2.u.b.161.1 4 100.67 even 20
400.2.u.b.241.1 4 100.47 even 20
625.2.a.b.1.2 2 5.2 odd 4
625.2.a.c.1.1 2 5.3 odd 4
625.2.b.a.624.2 4 1.1 even 1 trivial
625.2.b.a.624.3 4 5.4 even 2 inner
625.2.d.b.126.1 4 25.13 odd 20
625.2.d.b.501.1 4 25.23 odd 20
625.2.d.h.126.1 4 25.12 odd 20
625.2.d.h.501.1 4 25.2 odd 20
625.2.e.c.124.1 8 25.11 even 5
625.2.e.c.124.2 8 25.14 even 10
625.2.e.c.499.1 8 25.9 even 10
625.2.e.c.499.2 8 25.16 even 5
5625.2.a.d.1.2 2 15.8 even 4
5625.2.a.f.1.1 2 15.2 even 4
10000.2.a.c.1.2 2 20.7 even 4
10000.2.a.l.1.1 2 20.3 even 4