# Properties

 Label 625.2.b.a.624.1 Level $625$ Weight $2$ Character 625.624 Analytic conductor $4.991$ Analytic rank $0$ Dimension $4$ CM no Inner twists $2$

# Related objects

## 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: $$x^{4} + 3x^{2} + 1$$ 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 624.1 Root $$-1.61803i$$ of defining polynomial Character $$\chi$$ $$=$$ 625.624 Dual form 625.2.b.a.624.4

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-1.61803i q^{2} -1.00000i q^{3} -0.618034 q^{4} -1.61803 q^{6} +0.618034i q^{7} -2.23607i q^{8} +2.00000 q^{9} +O(q^{10})$$ $$q-1.61803i q^{2} -1.00000i q^{3} -0.618034 q^{4} -1.61803 q^{6} +0.618034i q^{7} -2.23607i q^{8} +2.00000 q^{9} -5.23607 q^{11} +0.618034i q^{12} -1.85410i q^{13} +1.00000 q^{14} -4.85410 q^{16} -5.23607i q^{17} -3.23607i q^{18} -0.854102 q^{19} +0.618034 q^{21} +8.47214i q^{22} -3.76393i q^{23} -2.23607 q^{24} -3.00000 q^{26} -5.00000i q^{27} -0.381966i q^{28} +3.61803 q^{29} -3.00000 q^{31} +3.38197i q^{32} +5.23607i q^{33} -8.47214 q^{34} -1.23607 q^{36} -0.236068i q^{37} +1.38197i q^{38} -1.85410 q^{39} -0.763932 q^{41} -1.00000i q^{42} +4.85410i q^{43} +3.23607 q^{44} -6.09017 q^{46} +0.618034i q^{47} +4.85410i q^{48} +6.61803 q^{49} -5.23607 q^{51} +1.14590i q^{52} +3.47214i q^{53} -8.09017 q^{54} +1.38197 q^{56} +0.854102i q^{57} -5.85410i q^{58} +10.8541 q^{59} +8.70820 q^{61} +4.85410i q^{62} +1.23607i q^{63} -4.23607 q^{64} +8.47214 q^{66} +4.76393i q^{67} +3.23607i q^{68} -3.76393 q^{69} -6.61803 q^{71} -4.47214i q^{72} +9.00000i q^{73} -0.381966 q^{74} +0.527864 q^{76} -3.23607i q^{77} +3.00000i q^{78} +8.09017 q^{79} +1.00000 q^{81} +1.23607i q^{82} +6.23607i q^{83} -0.381966 q^{84} +7.85410 q^{86} -3.61803i q^{87} +11.7082i q^{88} +8.94427 q^{89} +1.14590 q^{91} +2.32624i q^{92} +3.00000i q^{93} +1.00000 q^{94} +3.38197 q^{96} -3.85410i q^{97} -10.7082i q^{98} -10.4721 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q + 2 q^{4} - 2 q^{6} + 8 q^{9}+O(q^{10})$$ 4 * q + 2 * q^4 - 2 * q^6 + 8 * q^9 $$4 q + 2 q^{4} - 2 q^{6} + 8 q^{9} - 12 q^{11} + 4 q^{14} - 6 q^{16} + 10 q^{19} - 2 q^{21} - 12 q^{26} + 10 q^{29} - 12 q^{31} - 16 q^{34} + 4 q^{36} + 6 q^{39} - 12 q^{41} + 4 q^{44} - 2 q^{46} + 22 q^{49} - 12 q^{51} - 10 q^{54} + 10 q^{56} + 30 q^{59} + 8 q^{61} - 8 q^{64} + 16 q^{66} - 24 q^{69} - 22 q^{71} - 6 q^{74} + 20 q^{76} + 10 q^{79} + 4 q^{81} - 6 q^{84} + 18 q^{86} + 18 q^{91} + 4 q^{94} + 18 q^{96} - 24 q^{99}+O(q^{100})$$ 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

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 1.61803i − 1.14412i −0.820211 0.572061i $$-0.806144\pi$$
0.820211 0.572061i $$-0.193856\pi$$
$$3$$ − 1.00000i − 0.577350i −0.957427 0.288675i $$-0.906785\pi$$
0.957427 0.288675i $$-0.0932147\pi$$
$$4$$ −0.618034 −0.309017
$$5$$ 0 0
$$6$$ −1.61803 −0.660560
$$7$$ 0.618034i 0.233595i 0.993156 + 0.116797i $$0.0372628\pi$$
−0.993156 + 0.116797i $$0.962737\pi$$
$$8$$ − 2.23607i − 0.790569i
$$9$$ 2.00000 0.666667
$$10$$ 0 0
$$11$$ −5.23607 −1.57873 −0.789367 0.613922i $$-0.789591\pi$$
−0.789367 + 0.613922i $$0.789591\pi$$
$$12$$ 0.618034i 0.178411i
$$13$$ − 1.85410i − 0.514235i −0.966380 0.257118i $$-0.917227\pi$$
0.966380 0.257118i $$-0.0827728\pi$$
$$14$$ 1.00000 0.267261
$$15$$ 0 0
$$16$$ −4.85410 −1.21353
$$17$$ − 5.23607i − 1.26993i −0.772540 0.634967i $$-0.781014\pi$$
0.772540 0.634967i $$-0.218986\pi$$
$$18$$ − 3.23607i − 0.762749i
$$19$$ −0.854102 −0.195944 −0.0979722 0.995189i $$-0.531236\pi$$
−0.0979722 + 0.995189i $$0.531236\pi$$
$$20$$ 0 0
$$21$$ 0.618034 0.134866
$$22$$ 8.47214i 1.80627i
$$23$$ − 3.76393i − 0.784834i −0.919787 0.392417i $$-0.871639\pi$$
0.919787 0.392417i $$-0.128361\pi$$
$$24$$ −2.23607 −0.456435
$$25$$ 0 0
$$26$$ −3.00000 −0.588348
$$27$$ − 5.00000i − 0.962250i
$$28$$ − 0.381966i − 0.0721848i
$$29$$ 3.61803 0.671852 0.335926 0.941888i $$-0.390951\pi$$
0.335926 + 0.941888i $$0.390951\pi$$
$$30$$ 0 0
$$31$$ −3.00000 −0.538816 −0.269408 0.963026i $$-0.586828\pi$$
−0.269408 + 0.963026i $$0.586828\pi$$
$$32$$ 3.38197i 0.597853i
$$33$$ 5.23607i 0.911482i
$$34$$ −8.47214 −1.45296
$$35$$ 0 0
$$36$$ −1.23607 −0.206011
$$37$$ − 0.236068i − 0.0388093i −0.999812 0.0194047i $$-0.993823\pi$$
0.999812 0.0194047i $$-0.00617709\pi$$
$$38$$ 1.38197i 0.224184i
$$39$$ −1.85410 −0.296894
$$40$$ 0 0
$$41$$ −0.763932 −0.119306 −0.0596531 0.998219i $$-0.518999\pi$$
−0.0596531 + 0.998219i $$0.518999\pi$$
$$42$$ − 1.00000i − 0.154303i
$$43$$ 4.85410i 0.740244i 0.928983 + 0.370122i $$0.120684\pi$$
−0.928983 + 0.370122i $$0.879316\pi$$
$$44$$ 3.23607 0.487856
$$45$$ 0 0
$$46$$ −6.09017 −0.897947
$$47$$ 0.618034i 0.0901495i 0.998984 + 0.0450748i $$0.0143526\pi$$
−0.998984 + 0.0450748i $$0.985647\pi$$
$$48$$ 4.85410i 0.700629i
$$49$$ 6.61803 0.945433
$$50$$ 0 0
$$51$$ −5.23607 −0.733196
$$52$$ 1.14590i 0.158907i
$$53$$ 3.47214i 0.476935i 0.971151 + 0.238467i $$0.0766450\pi$$
−0.971151 + 0.238467i $$0.923355\pi$$
$$54$$ −8.09017 −1.10093
$$55$$ 0 0
$$56$$ 1.38197 0.184673
$$57$$ 0.854102i 0.113129i
$$58$$ − 5.85410i − 0.768681i
$$59$$ 10.8541 1.41308 0.706542 0.707671i $$-0.250254\pi$$
0.706542 + 0.707671i $$0.250254\pi$$
$$60$$ 0 0
$$61$$ 8.70820 1.11497 0.557486 0.830187i $$-0.311766\pi$$
0.557486 + 0.830187i $$0.311766\pi$$
$$62$$ 4.85410i 0.616472i
$$63$$ 1.23607i 0.155730i
$$64$$ −4.23607 −0.529508
$$65$$ 0 0
$$66$$ 8.47214 1.04285
$$67$$ 4.76393i 0.582007i 0.956722 + 0.291003i $$0.0939891\pi$$
−0.956722 + 0.291003i $$0.906011\pi$$
$$68$$ 3.23607i 0.392431i
$$69$$ −3.76393 −0.453124
$$70$$ 0 0
$$71$$ −6.61803 −0.785416 −0.392708 0.919663i $$-0.628462\pi$$
−0.392708 + 0.919663i $$0.628462\pi$$
$$72$$ − 4.47214i − 0.527046i
$$73$$ 9.00000i 1.05337i 0.850060 + 0.526685i $$0.176565\pi$$
−0.850060 + 0.526685i $$0.823435\pi$$
$$74$$ −0.381966 −0.0444026
$$75$$ 0 0
$$76$$ 0.527864 0.0605502
$$77$$ − 3.23607i − 0.368784i
$$78$$ 3.00000i 0.339683i
$$79$$ 8.09017 0.910215 0.455108 0.890436i $$-0.349601\pi$$
0.455108 + 0.890436i $$0.349601\pi$$
$$80$$ 0 0
$$81$$ 1.00000 0.111111
$$82$$ 1.23607i 0.136501i
$$83$$ 6.23607i 0.684497i 0.939609 + 0.342249i $$0.111189\pi$$
−0.939609 + 0.342249i $$0.888811\pi$$
$$84$$ −0.381966 −0.0416759
$$85$$ 0 0
$$86$$ 7.85410 0.846930
$$87$$ − 3.61803i − 0.387894i
$$88$$ 11.7082i 1.24810i
$$89$$ 8.94427 0.948091 0.474045 0.880500i $$-0.342793\pi$$
0.474045 + 0.880500i $$0.342793\pi$$
$$90$$ 0 0
$$91$$ 1.14590 0.120123
$$92$$ 2.32624i 0.242527i
$$93$$ 3.00000i 0.311086i
$$94$$ 1.00000 0.103142
$$95$$ 0 0
$$96$$ 3.38197 0.345170
$$97$$ − 3.85410i − 0.391325i −0.980671 0.195662i $$-0.937314\pi$$
0.980671 0.195662i $$-0.0626857\pi$$
$$98$$ − 10.7082i − 1.08169i
$$99$$ −10.4721 −1.05249
$$100$$ 0 0
$$101$$ 1.47214 0.146483 0.0732415 0.997314i $$-0.476666\pi$$
0.0732415 + 0.997314i $$0.476666\pi$$
$$102$$ 8.47214i 0.838866i
$$103$$ − 8.56231i − 0.843669i −0.906673 0.421835i $$-0.861386\pi$$
0.906673 0.421835i $$-0.138614\pi$$
$$104$$ −4.14590 −0.406539
$$105$$ 0 0
$$106$$ 5.61803 0.545672
$$107$$ − 16.4164i − 1.58703i −0.608548 0.793517i $$-0.708248\pi$$
0.608548 0.793517i $$-0.291752\pi$$
$$108$$ 3.09017i 0.297352i
$$109$$ −10.0000 −0.957826 −0.478913 0.877862i $$-0.658969\pi$$
−0.478913 + 0.877862i $$0.658969\pi$$
$$110$$ 0 0
$$111$$ −0.236068 −0.0224066
$$112$$ − 3.00000i − 0.283473i
$$113$$ − 16.8541i − 1.58550i −0.609547 0.792750i $$-0.708648\pi$$
0.609547 0.792750i $$-0.291352\pi$$
$$114$$ 1.38197 0.129433
$$115$$ 0 0
$$116$$ −2.23607 −0.207614
$$117$$ − 3.70820i − 0.342824i
$$118$$ − 17.5623i − 1.61674i
$$119$$ 3.23607 0.296650
$$120$$ 0 0
$$121$$ 16.4164 1.49240
$$122$$ − 14.0902i − 1.27566i
$$123$$ 0.763932i 0.0688814i
$$124$$ 1.85410 0.166503
$$125$$ 0 0
$$126$$ 2.00000 0.178174
$$127$$ 19.8885i 1.76482i 0.470479 + 0.882411i $$0.344081\pi$$
−0.470479 + 0.882411i $$0.655919\pi$$
$$128$$ 13.6180i 1.20368i
$$129$$ 4.85410 0.427380
$$130$$ 0 0
$$131$$ 6.79837 0.593977 0.296988 0.954881i $$-0.404018\pi$$
0.296988 + 0.954881i $$0.404018\pi$$
$$132$$ − 3.23607i − 0.281664i
$$133$$ − 0.527864i − 0.0457716i
$$134$$ 7.70820 0.665887
$$135$$ 0 0
$$136$$ −11.7082 −1.00397
$$137$$ − 11.9443i − 1.02047i −0.860036 0.510234i $$-0.829559\pi$$
0.860036 0.510234i $$-0.170441\pi$$
$$138$$ 6.09017i 0.518430i
$$139$$ −5.00000 −0.424094 −0.212047 0.977259i $$-0.568013\pi$$
−0.212047 + 0.977259i $$0.568013\pi$$
$$140$$ 0 0
$$141$$ 0.618034 0.0520479
$$142$$ 10.7082i 0.898613i
$$143$$ 9.70820i 0.811841i
$$144$$ −9.70820 −0.809017
$$145$$ 0 0
$$146$$ 14.5623 1.20519
$$147$$ − 6.61803i − 0.545846i
$$148$$ 0.145898i 0.0119927i
$$149$$ 3.94427 0.323127 0.161564 0.986862i $$-0.448346\pi$$
0.161564 + 0.986862i $$0.448346\pi$$
$$150$$ 0 0
$$151$$ 14.5623 1.18506 0.592532 0.805547i $$-0.298128\pi$$
0.592532 + 0.805547i $$0.298128\pi$$
$$152$$ 1.90983i 0.154908i
$$153$$ − 10.4721i − 0.846622i
$$154$$ −5.23607 −0.421934
$$155$$ 0 0
$$156$$ 1.14590 0.0917453
$$157$$ 13.1803i 1.05191i 0.850514 + 0.525953i $$0.176291\pi$$
−0.850514 + 0.525953i $$0.823709\pi$$
$$158$$ − 13.0902i − 1.04140i
$$159$$ 3.47214 0.275358
$$160$$ 0 0
$$161$$ 2.32624 0.183333
$$162$$ − 1.61803i − 0.127125i
$$163$$ − 11.0000i − 0.861586i −0.902451 0.430793i $$-0.858234\pi$$
0.902451 0.430793i $$-0.141766\pi$$
$$164$$ 0.472136 0.0368676
$$165$$ 0 0
$$166$$ 10.0902 0.783149
$$167$$ 14.5623i 1.12687i 0.826162 + 0.563433i $$0.190520\pi$$
−0.826162 + 0.563433i $$0.809480\pi$$
$$168$$ − 1.38197i − 0.106621i
$$169$$ 9.56231 0.735562
$$170$$ 0 0
$$171$$ −1.70820 −0.130630
$$172$$ − 3.00000i − 0.228748i
$$173$$ − 18.8885i − 1.43607i −0.696007 0.718035i $$-0.745042\pi$$
0.696007 0.718035i $$-0.254958\pi$$
$$174$$ −5.85410 −0.443798
$$175$$ 0 0
$$176$$ 25.4164 1.91583
$$177$$ − 10.8541i − 0.815844i
$$178$$ − 14.4721i − 1.08473i
$$179$$ 0.527864 0.0394544 0.0197272 0.999805i $$-0.493720\pi$$
0.0197272 + 0.999805i $$0.493720\pi$$
$$180$$ 0 0
$$181$$ 0.291796 0.0216890 0.0108445 0.999941i $$-0.496548\pi$$
0.0108445 + 0.999941i $$0.496548\pi$$
$$182$$ − 1.85410i − 0.137435i
$$183$$ − 8.70820i − 0.643729i
$$184$$ −8.41641 −0.620466
$$185$$ 0 0
$$186$$ 4.85410 0.355920
$$187$$ 27.4164i 2.00489i
$$188$$ − 0.381966i − 0.0278577i
$$189$$ 3.09017 0.224777
$$190$$ 0 0
$$191$$ −1.81966 −0.131666 −0.0658330 0.997831i $$-0.520970\pi$$
−0.0658330 + 0.997831i $$0.520970\pi$$
$$192$$ 4.23607i 0.305712i
$$193$$ − 7.70820i − 0.554849i −0.960747 0.277424i $$-0.910519\pi$$
0.960747 0.277424i $$-0.0894808\pi$$
$$194$$ −6.23607 −0.447724
$$195$$ 0 0
$$196$$ −4.09017 −0.292155
$$197$$ 3.70820i 0.264199i 0.991236 + 0.132099i $$0.0421718\pi$$
−0.991236 + 0.132099i $$0.957828\pi$$
$$198$$ 16.9443i 1.20418i
$$199$$ 17.5623 1.24496 0.622479 0.782636i $$-0.286125\pi$$
0.622479 + 0.782636i $$0.286125\pi$$
$$200$$ 0 0
$$201$$ 4.76393 0.336022
$$202$$ − 2.38197i − 0.167595i
$$203$$ 2.23607i 0.156941i
$$204$$ 3.23607 0.226570
$$205$$ 0 0
$$206$$ −13.8541 −0.965261
$$207$$ − 7.52786i − 0.523223i
$$208$$ 9.00000i 0.624038i
$$209$$ 4.47214 0.309344
$$210$$ 0 0
$$211$$ −9.18034 −0.632001 −0.316000 0.948759i $$-0.602340\pi$$
−0.316000 + 0.948759i $$0.602340\pi$$
$$212$$ − 2.14590i − 0.147381i
$$213$$ 6.61803i 0.453460i
$$214$$ −26.5623 −1.81576
$$215$$ 0 0
$$216$$ −11.1803 −0.760726
$$217$$ − 1.85410i − 0.125865i
$$218$$ 16.1803i 1.09587i
$$219$$ 9.00000 0.608164
$$220$$ 0 0
$$221$$ −9.70820 −0.653044
$$222$$ 0.381966i 0.0256359i
$$223$$ 0.180340i 0.0120765i 0.999982 + 0.00603823i $$0.00192204\pi$$
−0.999982 + 0.00603823i $$0.998078\pi$$
$$224$$ −2.09017 −0.139655
$$225$$ 0 0
$$226$$ −27.2705 −1.81401
$$227$$ 14.7639i 0.979917i 0.871746 + 0.489958i $$0.162988\pi$$
−0.871746 + 0.489958i $$0.837012\pi$$
$$228$$ − 0.527864i − 0.0349587i
$$229$$ −21.7082 −1.43452 −0.717259 0.696806i $$-0.754604\pi$$
−0.717259 + 0.696806i $$0.754604\pi$$
$$230$$ 0 0
$$231$$ −3.23607 −0.212918
$$232$$ − 8.09017i − 0.531146i
$$233$$ 2.94427i 0.192886i 0.995339 + 0.0964428i $$0.0307465\pi$$
−0.995339 + 0.0964428i $$0.969254\pi$$
$$234$$ −6.00000 −0.392232
$$235$$ 0 0
$$236$$ −6.70820 −0.436667
$$237$$ − 8.09017i − 0.525513i
$$238$$ − 5.23607i − 0.339404i
$$239$$ 20.5279 1.32784 0.663919 0.747805i $$-0.268892\pi$$
0.663919 + 0.747805i $$0.268892\pi$$
$$240$$ 0 0
$$241$$ 2.52786 0.162834 0.0814170 0.996680i $$-0.474055\pi$$
0.0814170 + 0.996680i $$0.474055\pi$$
$$242$$ − 26.5623i − 1.70749i
$$243$$ − 16.0000i − 1.02640i
$$244$$ −5.38197 −0.344545
$$245$$ 0 0
$$246$$ 1.23607 0.0788088
$$247$$ 1.58359i 0.100762i
$$248$$ 6.70820i 0.425971i
$$249$$ 6.23607 0.395195
$$250$$ 0 0
$$251$$ −29.1803 −1.84185 −0.920923 0.389744i $$-0.872564\pi$$
−0.920923 + 0.389744i $$0.872564\pi$$
$$252$$ − 0.763932i − 0.0481232i
$$253$$ 19.7082i 1.23904i
$$254$$ 32.1803 2.01917
$$255$$ 0 0
$$256$$ 13.5623 0.847644
$$257$$ 22.8541i 1.42560i 0.701367 + 0.712800i $$0.252573\pi$$
−0.701367 + 0.712800i $$0.747427\pi$$
$$258$$ − 7.85410i − 0.488975i
$$259$$ 0.145898 0.00906566
$$260$$ 0 0
$$261$$ 7.23607 0.447901
$$262$$ − 11.0000i − 0.679582i
$$263$$ 10.9098i 0.672729i 0.941732 + 0.336364i $$0.109197\pi$$
−0.941732 + 0.336364i $$0.890803\pi$$
$$264$$ 11.7082 0.720590
$$265$$ 0 0
$$266$$ −0.854102 −0.0523684
$$267$$ − 8.94427i − 0.547381i
$$268$$ − 2.94427i − 0.179850i
$$269$$ 12.7639 0.778231 0.389115 0.921189i $$-0.372781\pi$$
0.389115 + 0.921189i $$0.372781\pi$$
$$270$$ 0 0
$$271$$ −8.00000 −0.485965 −0.242983 0.970031i $$-0.578126\pi$$
−0.242983 + 0.970031i $$0.578126\pi$$
$$272$$ 25.4164i 1.54110i
$$273$$ − 1.14590i − 0.0693529i
$$274$$ −19.3262 −1.16754
$$275$$ 0 0
$$276$$ 2.32624 0.140023
$$277$$ − 24.7082i − 1.48457i −0.670083 0.742286i $$-0.733742\pi$$
0.670083 0.742286i $$-0.266258\pi$$
$$278$$ 8.09017i 0.485216i
$$279$$ −6.00000 −0.359211
$$280$$ 0 0
$$281$$ 10.0902 0.601929 0.300965 0.953635i $$-0.402691\pi$$
0.300965 + 0.953635i $$0.402691\pi$$
$$282$$ − 1.00000i − 0.0595491i
$$283$$ 29.8541i 1.77464i 0.461152 + 0.887321i $$0.347436\pi$$
−0.461152 + 0.887321i $$0.652564\pi$$
$$284$$ 4.09017 0.242707
$$285$$ 0 0
$$286$$ 15.7082 0.928846
$$287$$ − 0.472136i − 0.0278693i
$$288$$ 6.76393i 0.398569i
$$289$$ −10.4164 −0.612730
$$290$$ 0 0
$$291$$ −3.85410 −0.225931
$$292$$ − 5.56231i − 0.325509i
$$293$$ 19.5279i 1.14083i 0.821357 + 0.570415i $$0.193218\pi$$
−0.821357 + 0.570415i $$0.806782\pi$$
$$294$$ −10.7082 −0.624515
$$295$$ 0 0
$$296$$ −0.527864 −0.0306815
$$297$$ 26.1803i 1.51914i
$$298$$ − 6.38197i − 0.369697i
$$299$$ −6.97871 −0.403589
$$300$$ 0 0
$$301$$ −3.00000 −0.172917
$$302$$ − 23.5623i − 1.35586i
$$303$$ − 1.47214i − 0.0845720i
$$304$$ 4.14590 0.237784
$$305$$ 0 0
$$306$$ −16.9443 −0.968640
$$307$$ 9.23607i 0.527130i 0.964642 + 0.263565i $$0.0848984\pi$$
−0.964642 + 0.263565i $$0.915102\pi$$
$$308$$ 2.00000i 0.113961i
$$309$$ −8.56231 −0.487093
$$310$$ 0 0
$$311$$ 8.50658 0.482364 0.241182 0.970480i $$-0.422465\pi$$
0.241182 + 0.970480i $$0.422465\pi$$
$$312$$ 4.14590i 0.234715i
$$313$$ 16.7639i 0.947553i 0.880645 + 0.473777i $$0.157110\pi$$
−0.880645 + 0.473777i $$0.842890\pi$$
$$314$$ 21.3262 1.20351
$$315$$ 0 0
$$316$$ −5.00000 −0.281272
$$317$$ 7.65248i 0.429806i 0.976635 + 0.214903i $$0.0689435\pi$$
−0.976635 + 0.214903i $$0.931056\pi$$
$$318$$ − 5.61803i − 0.315044i
$$319$$ −18.9443 −1.06068
$$320$$ 0 0
$$321$$ −16.4164 −0.916275
$$322$$ − 3.76393i − 0.209756i
$$323$$ 4.47214i 0.248836i
$$324$$ −0.618034 −0.0343352
$$325$$ 0 0
$$326$$ −17.7984 −0.985761
$$327$$ 10.0000i 0.553001i
$$328$$ 1.70820i 0.0943198i
$$329$$ −0.381966 −0.0210585
$$330$$ 0 0
$$331$$ −23.1246 −1.27104 −0.635522 0.772083i $$-0.719215\pi$$
−0.635522 + 0.772083i $$0.719215\pi$$
$$332$$ − 3.85410i − 0.211521i
$$333$$ − 0.472136i − 0.0258729i
$$334$$ 23.5623 1.28927
$$335$$ 0 0
$$336$$ −3.00000 −0.163663
$$337$$ 7.85410i 0.427840i 0.976851 + 0.213920i $$0.0686232\pi$$
−0.976851 + 0.213920i $$0.931377\pi$$
$$338$$ − 15.4721i − 0.841573i
$$339$$ −16.8541 −0.915389
$$340$$ 0 0
$$341$$ 15.7082 0.850647
$$342$$ 2.76393i 0.149456i
$$343$$ 8.41641i 0.454443i
$$344$$ 10.8541 0.585214
$$345$$ 0 0
$$346$$ −30.5623 −1.64304
$$347$$ − 19.9098i − 1.06882i −0.845227 0.534408i $$-0.820535\pi$$
0.845227 0.534408i $$-0.179465\pi$$
$$348$$ 2.23607i 0.119866i
$$349$$ −21.7082 −1.16201 −0.581007 0.813899i $$-0.697341\pi$$
−0.581007 + 0.813899i $$0.697341\pi$$
$$350$$ 0 0
$$351$$ −9.27051 −0.494823
$$352$$ − 17.7082i − 0.943850i
$$353$$ − 12.9098i − 0.687121i −0.939131 0.343560i $$-0.888367\pi$$
0.939131 0.343560i $$-0.111633\pi$$
$$354$$ −17.5623 −0.933426
$$355$$ 0 0
$$356$$ −5.52786 −0.292976
$$357$$ − 3.23607i − 0.171271i
$$358$$ − 0.854102i − 0.0451407i
$$359$$ −13.7426 −0.725309 −0.362655 0.931924i $$-0.618130\pi$$
−0.362655 + 0.931924i $$0.618130\pi$$
$$360$$ 0 0
$$361$$ −18.2705 −0.961606
$$362$$ − 0.472136i − 0.0248149i
$$363$$ − 16.4164i − 0.861638i
$$364$$ −0.708204 −0.0371200
$$365$$ 0 0
$$366$$ −14.0902 −0.736505
$$367$$ − 25.5623i − 1.33434i −0.744905 0.667171i $$-0.767505\pi$$
0.744905 0.667171i $$-0.232495\pi$$
$$368$$ 18.2705i 0.952416i
$$369$$ −1.52786 −0.0795374
$$370$$ 0 0
$$371$$ −2.14590 −0.111409
$$372$$ − 1.85410i − 0.0961307i
$$373$$ 28.2705i 1.46379i 0.681417 + 0.731896i $$0.261364\pi$$
−0.681417 + 0.731896i $$0.738636\pi$$
$$374$$ 44.3607 2.29384
$$375$$ 0 0
$$376$$ 1.38197 0.0712695
$$377$$ − 6.70820i − 0.345490i
$$378$$ − 5.00000i − 0.257172i
$$379$$ −14.5967 −0.749785 −0.374892 0.927068i $$-0.622320\pi$$
−0.374892 + 0.927068i $$0.622320\pi$$
$$380$$ 0 0
$$381$$ 19.8885 1.01892
$$382$$ 2.94427i 0.150642i
$$383$$ − 33.3607i − 1.70465i −0.523012 0.852326i $$-0.675192\pi$$
0.523012 0.852326i $$-0.324808\pi$$
$$384$$ 13.6180 0.694942
$$385$$ 0 0
$$386$$ −12.4721 −0.634815
$$387$$ 9.70820i 0.493496i
$$388$$ 2.38197i 0.120926i
$$389$$ −15.0000 −0.760530 −0.380265 0.924878i $$-0.624167\pi$$
−0.380265 + 0.924878i $$0.624167\pi$$
$$390$$ 0 0
$$391$$ −19.7082 −0.996687
$$392$$ − 14.7984i − 0.747431i
$$393$$ − 6.79837i − 0.342933i
$$394$$ 6.00000 0.302276
$$395$$ 0 0
$$396$$ 6.47214 0.325237
$$397$$ 29.0344i 1.45720i 0.684941 + 0.728598i $$0.259828\pi$$
−0.684941 + 0.728598i $$0.740172\pi$$
$$398$$ − 28.4164i − 1.42439i
$$399$$ −0.527864 −0.0264263
$$400$$ 0 0
$$401$$ 26.5967 1.32818 0.664089 0.747653i $$-0.268820\pi$$
0.664089 + 0.747653i $$0.268820\pi$$
$$402$$ − 7.70820i − 0.384450i
$$403$$ 5.56231i 0.277078i
$$404$$ −0.909830 −0.0452657
$$405$$ 0 0
$$406$$ 3.61803 0.179560
$$407$$ 1.23607i 0.0612696i
$$408$$ 11.7082i 0.579642i
$$409$$ −1.58359 −0.0783036 −0.0391518 0.999233i $$-0.512466\pi$$
−0.0391518 + 0.999233i $$0.512466\pi$$
$$410$$ 0 0
$$411$$ −11.9443 −0.589167
$$412$$ 5.29180i 0.260708i
$$413$$ 6.70820i 0.330089i
$$414$$ −12.1803 −0.598631
$$415$$ 0 0
$$416$$ 6.27051 0.307437
$$417$$ 5.00000i 0.244851i
$$418$$ − 7.23607i − 0.353928i
$$419$$ 9.47214 0.462744 0.231372 0.972865i $$-0.425679\pi$$
0.231372 + 0.972865i $$0.425679\pi$$
$$420$$ 0 0
$$421$$ 32.0000 1.55958 0.779792 0.626038i $$-0.215325\pi$$
0.779792 + 0.626038i $$0.215325\pi$$
$$422$$ 14.8541i 0.723086i
$$423$$ 1.23607i 0.0600997i
$$424$$ 7.76393 0.377050
$$425$$ 0 0
$$426$$ 10.7082 0.518814
$$427$$ 5.38197i 0.260452i
$$428$$ 10.1459i 0.490420i
$$429$$ 9.70820 0.468717
$$430$$ 0 0
$$431$$ −29.8328 −1.43700 −0.718498 0.695529i $$-0.755170\pi$$
−0.718498 + 0.695529i $$0.755170\pi$$
$$432$$ 24.2705i 1.16772i
$$433$$ − 26.8541i − 1.29053i −0.763961 0.645263i $$-0.776748\pi$$
0.763961 0.645263i $$-0.223252\pi$$
$$434$$ −3.00000 −0.144005
$$435$$ 0 0
$$436$$ 6.18034 0.295985
$$437$$ 3.21478i 0.153784i
$$438$$ − 14.5623i − 0.695814i
$$439$$ 40.9787 1.95581 0.977904 0.209056i $$-0.0670391\pi$$
0.977904 + 0.209056i $$0.0670391\pi$$
$$440$$ 0 0
$$441$$ 13.2361 0.630289
$$442$$ 15.7082i 0.747163i
$$443$$ − 29.9443i − 1.42270i −0.702840 0.711348i $$-0.748085\pi$$
0.702840 0.711348i $$-0.251915\pi$$
$$444$$ 0.145898 0.00692401
$$445$$ 0 0
$$446$$ 0.291796 0.0138169
$$447$$ − 3.94427i − 0.186558i
$$448$$ − 2.61803i − 0.123690i
$$449$$ −4.67376 −0.220568 −0.110284 0.993900i $$-0.535176\pi$$
−0.110284 + 0.993900i $$0.535176\pi$$
$$450$$ 0 0
$$451$$ 4.00000 0.188353
$$452$$ 10.4164i 0.489947i
$$453$$ − 14.5623i − 0.684197i
$$454$$ 23.8885 1.12114
$$455$$ 0 0
$$456$$ 1.90983 0.0894360
$$457$$ − 21.4164i − 1.00182i −0.865500 0.500909i $$-0.832999\pi$$
0.865500 0.500909i $$-0.167001\pi$$
$$458$$ 35.1246i 1.64127i
$$459$$ −26.1803 −1.22199
$$460$$ 0 0
$$461$$ 0.819660 0.0381754 0.0190877 0.999818i $$-0.493924\pi$$
0.0190877 + 0.999818i $$0.493924\pi$$
$$462$$ 5.23607i 0.243604i
$$463$$ 24.1246i 1.12117i 0.828098 + 0.560583i $$0.189423\pi$$
−0.828098 + 0.560583i $$0.810577\pi$$
$$464$$ −17.5623 −0.815310
$$465$$ 0 0
$$466$$ 4.76393 0.220685
$$467$$ 27.4508i 1.27027i 0.772400 + 0.635137i $$0.219056\pi$$
−0.772400 + 0.635137i $$0.780944\pi$$
$$468$$ 2.29180i 0.105938i
$$469$$ −2.94427 −0.135954
$$470$$ 0 0
$$471$$ 13.1803 0.607318
$$472$$ − 24.2705i − 1.11714i
$$473$$ − 25.4164i − 1.16865i
$$474$$ −13.0902 −0.601251
$$475$$ 0 0
$$476$$ −2.00000 −0.0916698
$$477$$ 6.94427i 0.317956i
$$478$$ − 33.2148i − 1.51921i
$$479$$ 10.8541 0.495937 0.247968 0.968768i $$-0.420237\pi$$
0.247968 + 0.968768i $$0.420237\pi$$
$$480$$ 0 0
$$481$$ −0.437694 −0.0199571
$$482$$ − 4.09017i − 0.186302i
$$483$$ − 2.32624i − 0.105847i
$$484$$ −10.1459 −0.461177
$$485$$ 0 0
$$486$$ −25.8885 −1.17433
$$487$$ − 36.4164i − 1.65018i −0.564998 0.825092i $$-0.691123\pi$$
0.564998 0.825092i $$-0.308877\pi$$
$$488$$ − 19.4721i − 0.881462i
$$489$$ −11.0000 −0.497437
$$490$$ 0 0
$$491$$ −43.2492 −1.95181 −0.975905 0.218196i $$-0.929983\pi$$
−0.975905 + 0.218196i $$0.929983\pi$$
$$492$$ − 0.472136i − 0.0212855i
$$493$$ − 18.9443i − 0.853207i
$$494$$ 2.56231 0.115284
$$495$$ 0 0
$$496$$ 14.5623 0.653867
$$497$$ − 4.09017i − 0.183469i
$$498$$ − 10.0902i − 0.452151i
$$499$$ −7.56231 −0.338535 −0.169268 0.985570i $$-0.554140\pi$$
−0.169268 + 0.985570i $$0.554140\pi$$
$$500$$ 0 0
$$501$$ 14.5623 0.650596
$$502$$ 47.2148i 2.10730i
$$503$$ 37.4164i 1.66832i 0.551526 + 0.834158i $$0.314046\pi$$
−0.551526 + 0.834158i $$0.685954\pi$$
$$504$$ 2.76393 0.123115
$$505$$ 0 0
$$506$$ 31.8885 1.41762
$$507$$ − 9.56231i − 0.424677i
$$508$$ − 12.2918i − 0.545360i
$$509$$ 20.3262 0.900945 0.450472 0.892790i $$-0.351256\pi$$
0.450472 + 0.892790i $$0.351256\pi$$
$$510$$ 0 0
$$511$$ −5.56231 −0.246062
$$512$$ 5.29180i 0.233867i
$$513$$ 4.27051i 0.188548i
$$514$$ 36.9787 1.63106
$$515$$ 0 0
$$516$$ −3.00000 −0.132068
$$517$$ − 3.23607i − 0.142322i
$$518$$ − 0.236068i − 0.0103722i
$$519$$ −18.8885 −0.829115
$$520$$ 0 0
$$521$$ 29.3607 1.28631 0.643157 0.765734i $$-0.277624\pi$$
0.643157 + 0.765734i $$0.277624\pi$$
$$522$$ − 11.7082i − 0.512454i
$$523$$ 13.1459i 0.574830i 0.957806 + 0.287415i $$0.0927959\pi$$
−0.957806 + 0.287415i $$0.907204\pi$$
$$524$$ −4.20163 −0.183549
$$525$$ 0 0
$$526$$ 17.6525 0.769685
$$527$$ 15.7082i 0.684260i
$$528$$ − 25.4164i − 1.10611i
$$529$$ 8.83282 0.384035
$$530$$ 0 0
$$531$$ 21.7082 0.942056
$$532$$ 0.326238i 0.0141442i
$$533$$ 1.41641i 0.0613514i
$$534$$ −14.4721 −0.626271
$$535$$ 0 0
$$536$$ 10.6525 0.460117
$$537$$ − 0.527864i − 0.0227790i
$$538$$ − 20.6525i − 0.890391i
$$539$$ −34.6525 −1.49259
$$540$$ 0 0
$$541$$ 27.1246 1.16618 0.583089 0.812408i $$-0.301844\pi$$
0.583089 + 0.812408i $$0.301844\pi$$
$$542$$ 12.9443i 0.556004i
$$543$$ − 0.291796i − 0.0125222i
$$544$$ 17.7082 0.759233
$$545$$ 0 0
$$546$$ −1.85410 −0.0793482
$$547$$ − 21.2918i − 0.910371i −0.890397 0.455186i $$-0.849573\pi$$
0.890397 0.455186i $$-0.150427\pi$$
$$548$$ 7.38197i 0.315342i
$$549$$ 17.4164 0.743314
$$550$$ 0 0
$$551$$ −3.09017 −0.131646
$$552$$ 8.41641i 0.358226i
$$553$$ 5.00000i 0.212622i
$$554$$ −39.9787 −1.69853
$$555$$ 0 0
$$556$$ 3.09017 0.131052
$$557$$ 4.76393i 0.201854i 0.994894 + 0.100927i $$0.0321809\pi$$
−0.994894 + 0.100927i $$0.967819\pi$$
$$558$$ 9.70820i 0.410981i
$$559$$ 9.00000 0.380659
$$560$$ 0 0
$$561$$ 27.4164 1.15752
$$562$$ − 16.3262i − 0.688681i
$$563$$ − 7.38197i − 0.311113i −0.987827 0.155556i $$-0.950283\pi$$
0.987827 0.155556i $$-0.0497170\pi$$
$$564$$ −0.381966 −0.0160837
$$565$$ 0 0
$$566$$ 48.3050 2.03041
$$567$$ 0.618034i 0.0259550i
$$568$$ 14.7984i 0.620926i
$$569$$ −20.5279 −0.860573 −0.430286 0.902692i $$-0.641587\pi$$
−0.430286 + 0.902692i $$0.641587\pi$$
$$570$$ 0 0
$$571$$ −8.12461 −0.340004 −0.170002 0.985444i $$-0.554377\pi$$
−0.170002 + 0.985444i $$0.554377\pi$$
$$572$$ − 6.00000i − 0.250873i
$$573$$ 1.81966i 0.0760174i
$$574$$ −0.763932 −0.0318859
$$575$$ 0 0
$$576$$ −8.47214 −0.353006
$$577$$ − 33.7771i − 1.40616i −0.711111 0.703079i $$-0.751808\pi$$
0.711111 0.703079i $$-0.248192\pi$$
$$578$$ 16.8541i 0.701038i
$$579$$ −7.70820 −0.320342
$$580$$ 0 0
$$581$$ −3.85410 −0.159895
$$582$$ 6.23607i 0.258493i
$$583$$ − 18.1803i − 0.752953i
$$584$$ 20.1246 0.832762
$$585$$ 0 0
$$586$$ 31.5967 1.30525
$$587$$ 5.29180i 0.218416i 0.994019 + 0.109208i $$0.0348314\pi$$
−0.994019 + 0.109208i $$0.965169\pi$$
$$588$$ 4.09017i 0.168676i
$$589$$ 2.56231 0.105578
$$590$$ 0 0
$$591$$ 3.70820 0.152535
$$592$$ 1.14590i 0.0470961i
$$593$$ 10.9098i 0.448013i 0.974588 + 0.224007i $$0.0719137\pi$$
−0.974588 + 0.224007i $$0.928086\pi$$
$$594$$ 42.3607 1.73808
$$595$$ 0 0
$$596$$ −2.43769 −0.0998518
$$597$$ − 17.5623i − 0.718777i
$$598$$ 11.2918i 0.461756i
$$599$$ 9.47214 0.387021 0.193510 0.981098i $$-0.438013\pi$$
0.193510 + 0.981098i $$0.438013\pi$$
$$600$$ 0 0
$$601$$ 2.72949 0.111338 0.0556691 0.998449i $$-0.482271\pi$$
0.0556691 + 0.998449i $$0.482271\pi$$
$$602$$ 4.85410i 0.197838i
$$603$$ 9.52786i 0.388005i
$$604$$ −9.00000 −0.366205
$$605$$ 0 0
$$606$$ −2.38197 −0.0967608
$$607$$ − 35.5623i − 1.44343i −0.692191 0.721715i $$-0.743354\pi$$
0.692191 0.721715i $$-0.256646\pi$$
$$608$$ − 2.88854i − 0.117146i
$$609$$ 2.23607 0.0906100
$$610$$ 0 0
$$611$$ 1.14590 0.0463581
$$612$$ 6.47214i 0.261621i
$$613$$ 14.9787i 0.604985i 0.953152 + 0.302492i $$0.0978186\pi$$
−0.953152 + 0.302492i $$0.902181\pi$$
$$614$$ 14.9443 0.603102
$$615$$ 0 0
$$616$$ −7.23607 −0.291549
$$617$$ 14.2361i 0.573123i 0.958062 + 0.286561i $$0.0925122\pi$$
−0.958062 + 0.286561i $$0.907488\pi$$
$$618$$ 13.8541i 0.557294i
$$619$$ −30.5279 −1.22702 −0.613509 0.789688i $$-0.710243\pi$$
−0.613509 + 0.789688i $$0.710243\pi$$
$$620$$ 0 0
$$621$$ −18.8197 −0.755207
$$622$$ − 13.7639i − 0.551883i
$$623$$ 5.52786i 0.221469i
$$624$$ 9.00000 0.360288
$$625$$ 0 0
$$626$$ 27.1246 1.08412
$$627$$ − 4.47214i − 0.178600i
$$628$$ − 8.14590i − 0.325057i
$$629$$ −1.23607 −0.0492853
$$630$$ 0 0
$$631$$ −10.2361 −0.407491 −0.203746 0.979024i $$-0.565312\pi$$
−0.203746 + 0.979024i $$0.565312\pi$$
$$632$$ − 18.0902i − 0.719588i
$$633$$ 9.18034i 0.364886i
$$634$$ 12.3820 0.491751
$$635$$ 0 0
$$636$$ −2.14590 −0.0850904
$$637$$ − 12.2705i − 0.486175i
$$638$$ 30.6525i 1.21354i
$$639$$ −13.2361 −0.523611
$$640$$ 0 0
$$641$$ −1.09017 −0.0430591 −0.0215296 0.999768i $$-0.506854\pi$$
−0.0215296 + 0.999768i $$0.506854\pi$$
$$642$$ 26.5623i 1.04833i
$$643$$ 30.8328i 1.21593i 0.793965 + 0.607964i $$0.208013\pi$$
−0.793965 + 0.607964i $$0.791987\pi$$
$$644$$ −1.43769 −0.0566531
$$645$$ 0 0
$$646$$ 7.23607 0.284699
$$647$$ − 36.5410i − 1.43658i −0.695746 0.718288i $$-0.744926\pi$$
0.695746 0.718288i $$-0.255074\pi$$
$$648$$ − 2.23607i − 0.0878410i
$$649$$ −56.8328 −2.23088
$$650$$ 0 0
$$651$$ −1.85410 −0.0726680
$$652$$ 6.79837i 0.266245i
$$653$$ − 19.0902i − 0.747056i −0.927619 0.373528i $$-0.878148\pi$$
0.927619 0.373528i $$-0.121852\pi$$
$$654$$ 16.1803 0.632701
$$655$$ 0 0
$$656$$ 3.70820 0.144781
$$657$$ 18.0000i 0.702247i
$$658$$ 0.618034i 0.0240935i
$$659$$ −15.5279 −0.604880 −0.302440 0.953168i $$-0.597801\pi$$
−0.302440 + 0.953168i $$0.597801\pi$$
$$660$$ 0 0
$$661$$ 19.6869 0.765732 0.382866 0.923804i $$-0.374937\pi$$
0.382866 + 0.923804i $$0.374937\pi$$
$$662$$ 37.4164i 1.45423i
$$663$$ 9.70820i 0.377035i
$$664$$ 13.9443 0.541143
$$665$$ 0 0
$$666$$ −0.763932 −0.0296018
$$667$$ − 13.6180i − 0.527292i
$$668$$ − 9.00000i − 0.348220i
$$669$$ 0.180340 0.00697234
$$670$$ 0 0
$$671$$ −45.5967 −1.76024
$$672$$ 2.09017i 0.0806301i
$$673$$ − 12.1803i − 0.469518i −0.972054 0.234759i $$-0.924570\pi$$
0.972054 0.234759i $$-0.0754300\pi$$
$$674$$ 12.7082 0.489502
$$675$$ 0 0
$$676$$ −5.90983 −0.227301
$$677$$ 10.6180i 0.408084i 0.978962 + 0.204042i $$0.0654079\pi$$
−0.978962 + 0.204042i $$0.934592\pi$$
$$678$$ 27.2705i 1.04732i
$$679$$ 2.38197 0.0914115
$$680$$ 0 0
$$681$$ 14.7639 0.565755
$$682$$ − 25.4164i − 0.973245i
$$683$$ 13.4721i 0.515497i 0.966212 + 0.257748i $$0.0829806\pi$$
−0.966212 + 0.257748i $$0.917019\pi$$
$$684$$ 1.05573 0.0403668
$$685$$ 0 0
$$686$$ 13.6180 0.519939
$$687$$ 21.7082i 0.828220i
$$688$$ − 23.5623i − 0.898304i
$$689$$ 6.43769 0.245257
$$690$$ 0 0
$$691$$ 36.2705 1.37980 0.689898 0.723907i $$-0.257656\pi$$
0.689898 + 0.723907i $$0.257656\pi$$
$$692$$ 11.6738i 0.443770i
$$693$$ − 6.47214i − 0.245856i
$$694$$ −32.2148 −1.22286
$$695$$ 0 0
$$696$$ −8.09017 −0.306657
$$697$$ 4.00000i 0.151511i
$$698$$ 35.1246i 1.32949i
$$699$$ 2.94427 0.111363
$$700$$ 0 0
$$701$$ −41.0132 −1.54905 −0.774523 0.632546i $$-0.782010\pi$$
−0.774523 + 0.632546i $$0.782010\pi$$
$$702$$ 15.0000i 0.566139i
$$703$$ 0.201626i 0.00760447i
$$704$$ 22.1803 0.835953
$$705$$ 0 0
$$706$$ −20.8885 −0.786151
$$707$$ 0.909830i 0.0342177i
$$708$$ 6.70820i 0.252110i
$$709$$ 33.5410 1.25966 0.629830 0.776733i $$-0.283125\pi$$
0.629830 + 0.776733i $$0.283125\pi$$
$$710$$ 0 0
$$711$$ 16.1803 0.606810
$$712$$ − 20.0000i − 0.749532i
$$713$$ 11.2918i 0.422881i
$$714$$ −5.23607 −0.195955
$$715$$ 0 0
$$716$$ −0.326238 −0.0121921
$$717$$ − 20.5279i − 0.766627i
$$718$$ 22.2361i 0.829843i
$$719$$ 23.2918 0.868637 0.434319 0.900759i $$-0.356989\pi$$
0.434319 + 0.900759i $$0.356989\pi$$
$$720$$ 0 0
$$721$$ 5.29180 0.197077
$$722$$ 29.5623i 1.10020i
$$723$$ − 2.52786i − 0.0940123i
$$724$$ −0.180340 −0.00670228
$$725$$ 0 0
$$726$$ −26.5623 −0.985820
$$727$$ 24.5623i 0.910966i 0.890245 + 0.455483i $$0.150533\pi$$
−0.890245 + 0.455483i $$0.849467\pi$$
$$728$$ − 2.56231i − 0.0949654i
$$729$$ −13.0000 −0.481481
$$730$$ 0 0
$$731$$ 25.4164 0.940060
$$732$$ 5.38197i 0.198923i
$$733$$ 19.9787i 0.737931i 0.929443 + 0.368965i $$0.120288\pi$$
−0.929443 + 0.368965i $$0.879712\pi$$
$$734$$ −41.3607 −1.52665
$$735$$ 0 0
$$736$$ 12.7295 0.469215
$$737$$ − 24.9443i − 0.918834i
$$738$$ 2.47214i 0.0910006i
$$739$$ −15.9787 −0.587786 −0.293893 0.955838i $$-0.594951\pi$$
−0.293893 + 0.955838i $$0.594951\pi$$
$$740$$ 0 0
$$741$$ 1.58359 0.0581747
$$742$$ 3.47214i 0.127466i
$$743$$ − 28.3607i − 1.04045i −0.854029 0.520226i $$-0.825848\pi$$
0.854029 0.520226i $$-0.174152\pi$$
$$744$$ 6.70820 0.245935
$$745$$ 0 0
$$746$$ 45.7426 1.67476
$$747$$ 12.4721i 0.456332i
$$748$$ − 16.9443i − 0.619544i
$$749$$ 10.1459 0.370723
$$750$$ 0 0
$$751$$ −5.11146 −0.186520 −0.0932598 0.995642i $$-0.529729\pi$$
−0.0932598 + 0.995642i $$0.529729\pi$$
$$752$$ − 3.00000i − 0.109399i
$$753$$ 29.1803i 1.06339i
$$754$$ −10.8541 −0.395283
$$755$$ 0 0
$$756$$ −1.90983 −0.0694598
$$757$$ 30.4164i 1.10550i 0.833346 + 0.552752i $$0.186422\pi$$
−0.833346 + 0.552752i $$0.813578\pi$$
$$758$$ 23.6180i 0.857846i
$$759$$ 19.7082 0.715362
$$760$$ 0 0
$$761$$ −18.4508 −0.668843 −0.334421 0.942424i $$-0.608541\pi$$
−0.334421 + 0.942424i $$0.608541\pi$$
$$762$$ − 32.1803i − 1.16577i
$$763$$ − 6.18034i − 0.223743i
$$764$$ 1.12461 0.0406870
$$765$$ 0 0
$$766$$ −53.9787 −1.95033
$$767$$ − 20.1246i − 0.726658i
$$768$$ − 13.5623i − 0.489388i
$$769$$ −13.4164 −0.483808 −0.241904 0.970300i $$-0.577772\pi$$
−0.241904 + 0.970300i $$0.577772\pi$$
$$770$$ 0 0
$$771$$ 22.8541 0.823070
$$772$$ 4.76393i 0.171458i
$$773$$ 36.1591i 1.30055i 0.759699 + 0.650275i $$0.225346\pi$$
−0.759699 + 0.650275i $$0.774654\pi$$
$$774$$ 15.7082 0.564620
$$775$$ 0 0
$$776$$ −8.61803 −0.309369
$$777$$ − 0.145898i − 0.00523406i
$$778$$ 24.2705i 0.870140i
$$779$$ 0.652476 0.0233774
$$780$$ 0 0
$$781$$ 34.6525 1.23996
$$782$$ 31.8885i 1.14033i
$$783$$ − 18.0902i − 0.646490i
$$784$$ −32.1246 −1.14731
$$785$$ 0 0
$$786$$ −11.0000 −0.392357
$$787$$ − 11.8197i − 0.421325i −0.977559 0.210663i $$-0.932438\pi$$
0.977559 0.210663i $$-0.0675622\pi$$
$$788$$ − 2.29180i − 0.0816419i
$$789$$ 10.9098 0.388400
$$790$$ 0 0
$$791$$ 10.4164 0.370365
$$792$$ 23.4164i 0.832066i
$$793$$ − 16.1459i − 0.573358i
$$794$$ 46.9787 1.66721
$$795$$ 0 0
$$796$$ −10.8541 −0.384713
$$797$$ 9.76393i 0.345856i 0.984934 + 0.172928i $$0.0553228\pi$$
−0.984934 + 0.172928i $$0.944677\pi$$
$$798$$ 0.854102i 0.0302349i
$$799$$ 3.23607 0.114484
$$800$$ 0 0
$$801$$ 17.8885 0.632061
$$802$$ − 43.0344i − 1.51960i
$$803$$ − 47.1246i − 1.66299i
$$804$$ −2.94427 −0.103836
$$805$$ 0 0
$$806$$ 9.00000 0.317011
$$807$$ − 12.7639i − 0.449312i
$$808$$ − 3.29180i − 0.115805i
$$809$$ −30.9787 −1.08915 −0.544577 0.838711i $$-0.683310\pi$$
−0.544577 + 0.838711i $$0.683310\pi$$
$$810$$ 0 0
$$811$$ −14.7082 −0.516475 −0.258237 0.966081i $$-0.583142\pi$$
−0.258237 + 0.966081i $$0.583142\pi$$
$$812$$ − 1.38197i − 0.0484975i
$$813$$ 8.00000i 0.280572i
$$814$$ 2.00000 0.0701000
$$815$$ 0 0
$$816$$ 25.4164 0.889752
$$817$$ − 4.14590i − 0.145047i
$$818$$ 2.56231i 0.0895889i
$$819$$ 2.29180 0.0800818
$$820$$ 0 0
$$821$$ −40.6869 −1.41998 −0.709992 0.704210i $$-0.751301\pi$$
−0.709992 + 0.704210i $$0.751301\pi$$
$$822$$ 19.3262i 0.674080i
$$823$$ − 47.7082i − 1.66300i −0.555522 0.831502i $$-0.687482\pi$$
0.555522 0.831502i $$-0.312518\pi$$
$$824$$ −19.1459 −0.666979
$$825$$ 0 0
$$826$$ 10.8541 0.377663
$$827$$ − 0.965558i − 0.0335757i −0.999859 0.0167879i $$-0.994656\pi$$
0.999859 0.0167879i $$-0.00534400\pi$$
$$828$$ 4.65248i 0.161685i
$$829$$ 35.8541 1.24526 0.622632 0.782515i $$-0.286063\pi$$
0.622632 + 0.782515i $$0.286063\pi$$
$$830$$ 0 0
$$831$$ −24.7082 −0.857118
$$832$$ 7.85410i 0.272292i
$$833$$ − 34.6525i − 1.20064i
$$834$$ 8.09017 0.280140
$$835$$ 0 0
$$836$$ −2.76393 −0.0955926
$$837$$ 15.0000i 0.518476i
$$838$$ − 15.3262i − 0.529436i
$$839$$ −10.8541 −0.374725 −0.187363 0.982291i $$-0.559994\pi$$
−0.187363 + 0.982291i $$0.559994\pi$$
$$840$$ 0 0
$$841$$ −15.9098 −0.548615
$$842$$ − 51.7771i − 1.78436i
$$843$$ − 10.0902i − 0.347524i
$$844$$ 5.67376 0.195299
$$845$$ 0 0
$$846$$ 2.00000 0.0687614
$$847$$ 10.1459i 0.348617i
$$848$$ − 16.8541i − 0.578772i
$$849$$ 29.8541 1.02459
$$850$$ 0 0
$$851$$ −0.888544 −0.0304589
$$852$$ − 4.09017i − 0.140127i
$$853$$ 15.3050i 0.524032i 0.965064 + 0.262016i $$0.0843873\pi$$
−0.965064 + 0.262016i $$0.915613\pi$$
$$854$$ 8.70820 0.297989
$$855$$ 0 0
$$856$$ −36.7082 −1.25466
$$857$$ 19.6869i 0.672492i 0.941774 + 0.336246i $$0.109157\pi$$
−0.941774 + 0.336246i $$0.890843\pi$$
$$858$$ − 15.7082i − 0.536269i
$$859$$ 1.58359 0.0540315 0.0270157 0.999635i $$-0.491400\pi$$
0.0270157 + 0.999635i $$0.491400\pi$$
$$860$$ 0 0
$$861$$ −0.472136 −0.0160904
$$862$$ 48.2705i 1.64410i
$$863$$ 21.4377i 0.729748i 0.931057 + 0.364874i $$0.118888\pi$$
−0.931057 + 0.364874i $$0.881112\pi$$
$$864$$ 16.9098 0.575284
$$865$$ 0 0
$$866$$ −43.4508 −1.47652
$$867$$ 10.4164i 0.353760i
$$868$$ 1.14590i 0.0388943i
$$869$$ −42.3607 −1.43699
$$870$$ 0 0
$$871$$ 8.83282 0.299289
$$872$$ 22.3607i 0.757228i
$$873$$ − 7.70820i − 0.260883i
$$874$$ 5.20163 0.175948
$$875$$ 0 0
$$876$$ −5.56231 −0.187933
$$877$$ − 36.5410i − 1.23390i −0.787001 0.616951i $$-0.788368\pi$$
0.787001 0.616951i $$-0.211632\pi$$
$$878$$ − 66.3050i − 2.23768i
$$879$$ 19.5279 0.658659
$$880$$ 0 0
$$881$$ −40.3607 −1.35979 −0.679893 0.733311i $$-0.737974\pi$$
−0.679893 + 0.733311i $$0.737974\pi$$
$$882$$ − 21.4164i − 0.721128i
$$883$$ 20.5836i 0.692693i 0.938107 + 0.346347i $$0.112578\pi$$
−0.938107 + 0.346347i $$0.887422\pi$$
$$884$$ 6.00000 0.201802
$$885$$ 0 0
$$886$$ −48.4508 −1.62774
$$887$$ 29.8885i 1.00356i 0.864996 + 0.501780i $$0.167321\pi$$
−0.864996 + 0.501780i $$0.832679\pi$$
$$888$$ 0.527864i 0.0177140i
$$889$$ −12.2918 −0.412254
$$890$$ 0 0
$$891$$ −5.23607 −0.175415
$$892$$ − 0.111456i − 0.00373183i
$$893$$ − 0.527864i − 0.0176643i
$$894$$ −6.38197 −0.213445
$$895$$ 0 0
$$896$$ −8.41641 −0.281172
$$897$$ 6.97871i 0.233012i
$$898$$ 7.56231i 0.252357i
$$899$$ −10.8541 −0.362005
$$900$$ 0 0
$$901$$ 18.1803 0.605675
$$902$$ − 6.47214i − 0.215499i
$$903$$ 3.00000i 0.0998337i
$$904$$ −37.6869 −1.25345
$$905$$ 0 0
$$906$$ −23.5623 −0.782805
$$907$$ − 33.2492i − 1.10402i −0.833837 0.552011i $$-0.813861\pi$$
0.833837 0.552011i $$-0.186139\pi$$
$$908$$ − 9.12461i − 0.302811i
$$909$$ 2.94427 0.0976553
$$910$$ 0 0
$$911$$ −40.2361 −1.33308 −0.666540 0.745469i $$-0.732226\pi$$
−0.666540 + 0.745469i $$0.732226\pi$$
$$912$$ − 4.14590i − 0.137284i
$$913$$ − 32.6525i − 1.08064i
$$914$$ −34.6525 −1.14620
$$915$$ 0 0
$$916$$ 13.4164 0.443291
$$917$$ 4.20163i 0.138750i
$$918$$ 42.3607i 1.39811i
$$919$$ 53.2148 1.75539 0.877697 0.479216i $$-0.159079\pi$$
0.877697 + 0.479216i $$0.159079\pi$$
$$920$$ 0 0
$$921$$ 9.23607 0.304339
$$922$$ − 1.32624i − 0.0436773i
$$923$$ 12.2705i 0.403889i
$$924$$ 2.00000 0.0657952
$$925$$ 0 0
$$926$$ 39.0344 1.28275
$$927$$ − 17.1246i − 0.562446i
$$928$$ 12.2361i 0.401669i
$$929$$ −41.6312 −1.36588 −0.682938 0.730477i $$-0.739298\pi$$
−0.682938 + 0.730477i $$0.739298\pi$$
$$930$$ 0 0
$$931$$ −5.65248 −0.185252
$$932$$ − 1.81966i − 0.0596049i
$$933$$ − 8.50658i − 0.278493i
$$934$$ 44.4164 1.45335
$$935$$ 0 0
$$936$$ −8.29180 −0.271026
$$937$$ 17.7295i 0.579197i 0.957148 + 0.289599i $$0.0935218\pi$$
−0.957148 + 0.289599i $$0.906478\pi$$
$$938$$ 4.76393i 0.155548i
$$939$$ 16.7639 0.547070
$$940$$ 0 0
$$941$$ −46.4164 −1.51313 −0.756566 0.653918i $$-0.773124\pi$$
−0.756566 + 0.653918i $$0.773124\pi$$
$$942$$ − 21.3262i − 0.694846i
$$943$$ 2.87539i 0.0936355i
$$944$$ −52.6869 −1.71481
$$945$$ 0 0
$$946$$ −41.1246 −1.33708
$$947$$ 2.65248i 0.0861939i 0.999071 + 0.0430969i $$0.0137224\pi$$
−0.999071 + 0.0430969i $$0.986278\pi$$
$$948$$ 5.00000i 0.162392i
$$949$$ 16.6869 0.541680
$$950$$ 0 0
$$951$$ 7.65248 0.248149
$$952$$ − 7.23607i − 0.234522i
$$953$$ 7.74265i 0.250809i 0.992106 + 0.125404i $$0.0400228\pi$$
−0.992106 + 0.125404i $$0.959977\pi$$
$$954$$ 11.2361 0.363781
$$955$$ 0 0
$$956$$ −12.6869 −0.410324
$$957$$ 18.9443i 0.612381i
$$958$$ − 17.5623i − 0.567412i
$$959$$ 7.38197 0.238376
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ 0.708204i 0.0228334i
$$963$$ − 32.8328i − 1.05802i
$$964$$ −1.56231 −0.0503185
$$965$$ 0 0
$$966$$ −3.76393 −0.121103
$$967$$ 4.11146i 0.132216i 0.997812 + 0.0661078i $$0.0210581\pi$$
−0.997812 + 0.0661078i $$0.978942\pi$$
$$968$$ − 36.7082i − 1.17985i
$$969$$ 4.47214 0.143666
$$970$$ 0 0
$$971$$ 5.61803 0.180291 0.0901456 0.995929i $$-0.471267\pi$$
0.0901456 + 0.995929i $$0.471267\pi$$
$$972$$ 9.88854i 0.317175i
$$973$$ − 3.09017i − 0.0990663i
$$974$$ −58.9230 −1.88801
$$975$$ 0 0
$$976$$ −42.2705 −1.35305
$$977$$ − 2.34752i − 0.0751040i −0.999295 0.0375520i $$-0.988044\pi$$
0.999295 0.0375520i $$-0.0119560\pi$$
$$978$$ 17.7984i 0.569129i
$$979$$ −46.8328 −1.49678
$$980$$ 0 0
$$981$$ −20.0000 −0.638551
$$982$$ 69.9787i 2.23311i
$$983$$ − 9.61803i − 0.306768i −0.988167 0.153384i $$-0.950983\pi$$
0.988167 0.153384i $$-0.0490171\pi$$
$$984$$ 1.70820 0.0544556
$$985$$ 0 0
$$986$$ −30.6525 −0.976174
$$987$$ 0.381966i 0.0121581i
$$988$$ − 0.978714i − 0.0311370i
$$989$$ 18.2705 0.580968
$$990$$ 0 0
$$991$$ −15.3607 −0.487948 −0.243974 0.969782i $$-0.578451\pi$$
−0.243974 + 0.969782i $$0.578451\pi$$
$$992$$ − 10.1459i − 0.322133i
$$993$$ 23.1246i 0.733837i
$$994$$ −6.61803 −0.209911
$$995$$ 0 0
$$996$$ −3.85410 −0.122122
$$997$$ 24.8885i 0.788228i 0.919062 + 0.394114i $$0.128949\pi$$
−0.919062 + 0.394114i $$0.871051\pi$$
$$998$$ 12.2361i 0.387326i
$$999$$ −1.18034 −0.0373443
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.624.1 4
5.2 odd 4 625.2.a.c.1.2 2
5.3 odd 4 625.2.a.b.1.1 2
5.4 even 2 inner 625.2.b.a.624.4 4
15.2 even 4 5625.2.a.d.1.1 2
15.8 even 4 5625.2.a.f.1.2 2
20.3 even 4 10000.2.a.c.1.1 2
20.7 even 4 10000.2.a.l.1.2 2
25.2 odd 20 125.2.d.a.101.1 4
25.3 odd 20 625.2.d.h.376.1 4
25.4 even 10 625.2.e.c.249.2 8
25.6 even 5 625.2.e.c.374.2 8
25.8 odd 20 625.2.d.h.251.1 4
25.9 even 10 125.2.e.a.99.1 8
25.11 even 5 125.2.e.a.24.1 8
25.12 odd 20 125.2.d.a.26.1 4
25.13 odd 20 25.2.d.a.6.1 4
25.14 even 10 125.2.e.a.24.2 8
25.16 even 5 125.2.e.a.99.2 8
25.17 odd 20 625.2.d.b.251.1 4
25.19 even 10 625.2.e.c.374.1 8
25.21 even 5 625.2.e.c.249.1 8
25.22 odd 20 625.2.d.b.376.1 4
25.23 odd 20 25.2.d.a.21.1 yes 4
75.23 even 20 225.2.h.b.46.1 4
75.38 even 20 225.2.h.b.181.1 4
100.23 even 20 400.2.u.b.321.1 4
100.63 even 20 400.2.u.b.81.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
25.2.d.a.6.1 4 25.13 odd 20
25.2.d.a.21.1 yes 4 25.23 odd 20
125.2.d.a.26.1 4 25.12 odd 20
125.2.d.a.101.1 4 25.2 odd 20
125.2.e.a.24.1 8 25.11 even 5
125.2.e.a.24.2 8 25.14 even 10
125.2.e.a.99.1 8 25.9 even 10
125.2.e.a.99.2 8 25.16 even 5
225.2.h.b.46.1 4 75.23 even 20
225.2.h.b.181.1 4 75.38 even 20
400.2.u.b.81.1 4 100.63 even 20
400.2.u.b.321.1 4 100.23 even 20
625.2.a.b.1.1 2 5.3 odd 4
625.2.a.c.1.2 2 5.2 odd 4
625.2.b.a.624.1 4 1.1 even 1 trivial
625.2.b.a.624.4 4 5.4 even 2 inner
625.2.d.b.251.1 4 25.17 odd 20
625.2.d.b.376.1 4 25.22 odd 20
625.2.d.h.251.1 4 25.8 odd 20
625.2.d.h.376.1 4 25.3 odd 20
625.2.e.c.249.1 8 25.21 even 5
625.2.e.c.249.2 8 25.4 even 10
625.2.e.c.374.1 8 25.19 even 10
625.2.e.c.374.2 8 25.6 even 5
5625.2.a.d.1.1 2 15.2 even 4
5625.2.a.f.1.2 2 15.8 even 4
10000.2.a.c.1.1 2 20.3 even 4
10000.2.a.l.1.2 2 20.7 even 4