# Properties

 Label 10000.2.a.l.1.2 Level $10000$ Weight $2$ Character 10000.1 Self dual yes Analytic conductor $79.850$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$10000 = 2^{4} \cdot 5^{4}$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 10000.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$79.8504020213$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\zeta_{10})^+$$ Defining polynomial: $$x^{2} - x - 1$$ x^2 - x - 1 Coefficient ring: $$\Z[a_1, \ldots, a_{7}]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 25) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.2 Root $$1.61803$$ of defining polynomial Character $$\chi$$ $$=$$ 10000.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q+1.00000 q^{3} +0.618034 q^{7} -2.00000 q^{9} +O(q^{10})$$ $$q+1.00000 q^{3} +0.618034 q^{7} -2.00000 q^{9} +5.23607 q^{11} -1.85410 q^{13} +5.23607 q^{17} -0.854102 q^{19} +0.618034 q^{21} +3.76393 q^{23} -5.00000 q^{27} -3.61803 q^{29} +3.00000 q^{31} +5.23607 q^{33} +0.236068 q^{37} -1.85410 q^{39} -0.763932 q^{41} -4.85410 q^{43} +0.618034 q^{47} -6.61803 q^{49} +5.23607 q^{51} +3.47214 q^{53} -0.854102 q^{57} +10.8541 q^{59} +8.70820 q^{61} -1.23607 q^{63} +4.76393 q^{67} +3.76393 q^{69} +6.61803 q^{71} +9.00000 q^{73} +3.23607 q^{77} +8.09017 q^{79} +1.00000 q^{81} -6.23607 q^{83} -3.61803 q^{87} -8.94427 q^{89} -1.14590 q^{91} +3.00000 q^{93} +3.85410 q^{97} -10.4721 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 2 q^{3} - q^{7} - 4 q^{9}+O(q^{10})$$ 2 * q + 2 * q^3 - q^7 - 4 * q^9 $$2 q + 2 q^{3} - q^{7} - 4 q^{9} + 6 q^{11} + 3 q^{13} + 6 q^{17} + 5 q^{19} - q^{21} + 12 q^{23} - 10 q^{27} - 5 q^{29} + 6 q^{31} + 6 q^{33} - 4 q^{37} + 3 q^{39} - 6 q^{41} - 3 q^{43} - q^{47} - 11 q^{49} + 6 q^{51} - 2 q^{53} + 5 q^{57} + 15 q^{59} + 4 q^{61} + 2 q^{63} + 14 q^{67} + 12 q^{69} + 11 q^{71} + 18 q^{73} + 2 q^{77} + 5 q^{79} + 2 q^{81} - 8 q^{83} - 5 q^{87} - 9 q^{91} + 6 q^{93} + q^{97} - 12 q^{99}+O(q^{100})$$ 2 * q + 2 * q^3 - q^7 - 4 * q^9 + 6 * q^11 + 3 * q^13 + 6 * q^17 + 5 * q^19 - q^21 + 12 * q^23 - 10 * q^27 - 5 * q^29 + 6 * q^31 + 6 * q^33 - 4 * q^37 + 3 * q^39 - 6 * q^41 - 3 * q^43 - q^47 - 11 * q^49 + 6 * q^51 - 2 * q^53 + 5 * q^57 + 15 * q^59 + 4 * q^61 + 2 * q^63 + 14 * q^67 + 12 * q^69 + 11 * q^71 + 18 * q^73 + 2 * q^77 + 5 * q^79 + 2 * q^81 - 8 * q^83 - 5 * q^87 - 9 * q^91 + 6 * q^93 + q^97 - 12 * q^99

## 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 0
$$3$$ 1.00000 0.577350 0.288675 0.957427i $$-0.406785\pi$$
0.288675 + 0.957427i $$0.406785\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 0.618034 0.233595 0.116797 0.993156i $$-0.462737\pi$$
0.116797 + 0.993156i $$0.462737\pi$$
$$8$$ 0 0
$$9$$ −2.00000 −0.666667
$$10$$ 0 0
$$11$$ 5.23607 1.57873 0.789367 0.613922i $$-0.210409\pi$$
0.789367 + 0.613922i $$0.210409\pi$$
$$12$$ 0 0
$$13$$ −1.85410 −0.514235 −0.257118 0.966380i $$-0.582773\pi$$
−0.257118 + 0.966380i $$0.582773\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ 5.23607 1.26993 0.634967 0.772540i $$-0.281014\pi$$
0.634967 + 0.772540i $$0.281014\pi$$
$$18$$ 0 0
$$19$$ −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
$$22$$ 0 0
$$23$$ 3.76393 0.784834 0.392417 0.919787i $$-0.371639\pi$$
0.392417 + 0.919787i $$0.371639\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ −5.00000 −0.962250
$$28$$ 0 0
$$29$$ −3.61803 −0.671852 −0.335926 0.941888i $$-0.609049\pi$$
−0.335926 + 0.941888i $$0.609049\pi$$
$$30$$ 0 0
$$31$$ 3.00000 0.538816 0.269408 0.963026i $$-0.413172\pi$$
0.269408 + 0.963026i $$0.413172\pi$$
$$32$$ 0 0
$$33$$ 5.23607 0.911482
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 0.236068 0.0388093 0.0194047 0.999812i $$-0.493823\pi$$
0.0194047 + 0.999812i $$0.493823\pi$$
$$38$$ 0 0
$$39$$ −1.85410 −0.296894
$$40$$ 0 0
$$41$$ −0.763932 −0.119306 −0.0596531 0.998219i $$-0.518999\pi$$
−0.0596531 + 0.998219i $$0.518999\pi$$
$$42$$ 0 0
$$43$$ −4.85410 −0.740244 −0.370122 0.928983i $$-0.620684\pi$$
−0.370122 + 0.928983i $$0.620684\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 0.618034 0.0901495 0.0450748 0.998984i $$-0.485647\pi$$
0.0450748 + 0.998984i $$0.485647\pi$$
$$48$$ 0 0
$$49$$ −6.61803 −0.945433
$$50$$ 0 0
$$51$$ 5.23607 0.733196
$$52$$ 0 0
$$53$$ 3.47214 0.476935 0.238467 0.971151i $$-0.423355\pi$$
0.238467 + 0.971151i $$0.423355\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −0.854102 −0.113129
$$58$$ 0 0
$$59$$ 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 $$0.311766\pi$$
$$62$$ 0 0
$$63$$ −1.23607 −0.155730
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 4.76393 0.582007 0.291003 0.956722i $$-0.406011\pi$$
0.291003 + 0.956722i $$0.406011\pi$$
$$68$$ 0 0
$$69$$ 3.76393 0.453124
$$70$$ 0 0
$$71$$ 6.61803 0.785416 0.392708 0.919663i $$-0.371538\pi$$
0.392708 + 0.919663i $$0.371538\pi$$
$$72$$ 0 0
$$73$$ 9.00000 1.05337 0.526685 0.850060i $$-0.323435\pi$$
0.526685 + 0.850060i $$0.323435\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 3.23607 0.368784
$$78$$ 0 0
$$79$$ 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
$$82$$ 0 0
$$83$$ −6.23607 −0.684497 −0.342249 0.939609i $$-0.611189\pi$$
−0.342249 + 0.939609i $$0.611189\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ −3.61803 −0.387894
$$88$$ 0 0
$$89$$ −8.94427 −0.948091 −0.474045 0.880500i $$-0.657207\pi$$
−0.474045 + 0.880500i $$0.657207\pi$$
$$90$$ 0 0
$$91$$ −1.14590 −0.120123
$$92$$ 0 0
$$93$$ 3.00000 0.311086
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 3.85410 0.391325 0.195662 0.980671i $$-0.437314\pi$$
0.195662 + 0.980671i $$0.437314\pi$$
$$98$$ 0 0
$$99$$ −10.4721 −1.05249
$$100$$ 0 0
$$101$$ 1.47214 0.146483 0.0732415 0.997314i $$-0.476666\pi$$
0.0732415 + 0.997314i $$0.476666\pi$$
$$102$$ 0 0
$$103$$ 8.56231 0.843669 0.421835 0.906673i $$-0.361386\pi$$
0.421835 + 0.906673i $$0.361386\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −16.4164 −1.58703 −0.793517 0.608548i $$-0.791752\pi$$
−0.793517 + 0.608548i $$0.791752\pi$$
$$108$$ 0 0
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ 0 0
$$111$$ 0.236068 0.0224066
$$112$$ 0 0
$$113$$ −16.8541 −1.58550 −0.792750 0.609547i $$-0.791352\pi$$
−0.792750 + 0.609547i $$0.791352\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 3.70820 0.342824
$$118$$ 0 0
$$119$$ 3.23607 0.296650
$$120$$ 0 0
$$121$$ 16.4164 1.49240
$$122$$ 0 0
$$123$$ −0.763932 −0.0688814
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 19.8885 1.76482 0.882411 0.470479i $$-0.155919\pi$$
0.882411 + 0.470479i $$0.155919\pi$$
$$128$$ 0 0
$$129$$ −4.85410 −0.427380
$$130$$ 0 0
$$131$$ −6.79837 −0.593977 −0.296988 0.954881i $$-0.595982\pi$$
−0.296988 + 0.954881i $$0.595982\pi$$
$$132$$ 0 0
$$133$$ −0.527864 −0.0457716
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 11.9443 1.02047 0.510234 0.860036i $$-0.329559\pi$$
0.510234 + 0.860036i $$0.329559\pi$$
$$138$$ 0 0
$$139$$ −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
$$142$$ 0 0
$$143$$ −9.70820 −0.811841
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ −6.61803 −0.545846
$$148$$ 0 0
$$149$$ −3.94427 −0.323127 −0.161564 0.986862i $$-0.551654\pi$$
−0.161564 + 0.986862i $$0.551654\pi$$
$$150$$ 0 0
$$151$$ −14.5623 −1.18506 −0.592532 0.805547i $$-0.701872\pi$$
−0.592532 + 0.805547i $$0.701872\pi$$
$$152$$ 0 0
$$153$$ −10.4721 −0.846622
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −13.1803 −1.05191 −0.525953 0.850514i $$-0.676291\pi$$
−0.525953 + 0.850514i $$0.676291\pi$$
$$158$$ 0 0
$$159$$ 3.47214 0.275358
$$160$$ 0 0
$$161$$ 2.32624 0.183333
$$162$$ 0 0
$$163$$ 11.0000 0.861586 0.430793 0.902451i $$-0.358234\pi$$
0.430793 + 0.902451i $$0.358234\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 14.5623 1.12687 0.563433 0.826162i $$-0.309480\pi$$
0.563433 + 0.826162i $$0.309480\pi$$
$$168$$ 0 0
$$169$$ −9.56231 −0.735562
$$170$$ 0 0
$$171$$ 1.70820 0.130630
$$172$$ 0 0
$$173$$ −18.8885 −1.43607 −0.718035 0.696007i $$-0.754958\pi$$
−0.718035 + 0.696007i $$0.754958\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ 10.8541 0.815844
$$178$$ 0 0
$$179$$ 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 $$0.496548\pi$$
$$182$$ 0 0
$$183$$ 8.70820 0.643729
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 27.4164 2.00489
$$188$$ 0 0
$$189$$ −3.09017 −0.224777
$$190$$ 0 0
$$191$$ 1.81966 0.131666 0.0658330 0.997831i $$-0.479030\pi$$
0.0658330 + 0.997831i $$0.479030\pi$$
$$192$$ 0 0
$$193$$ −7.70820 −0.554849 −0.277424 0.960747i $$-0.589481\pi$$
−0.277424 + 0.960747i $$0.589481\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −3.70820 −0.264199 −0.132099 0.991236i $$-0.542172\pi$$
−0.132099 + 0.991236i $$0.542172\pi$$
$$198$$ 0 0
$$199$$ 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
$$202$$ 0 0
$$203$$ −2.23607 −0.156941
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ −7.52786 −0.523223
$$208$$ 0 0
$$209$$ −4.47214 −0.309344
$$210$$ 0 0
$$211$$ 9.18034 0.632001 0.316000 0.948759i $$-0.397660\pi$$
0.316000 + 0.948759i $$0.397660\pi$$
$$212$$ 0 0
$$213$$ 6.61803 0.453460
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 1.85410 0.125865
$$218$$ 0 0
$$219$$ 9.00000 0.608164
$$220$$ 0 0
$$221$$ −9.70820 −0.653044
$$222$$ 0 0
$$223$$ −0.180340 −0.0120765 −0.00603823 0.999982i $$-0.501922\pi$$
−0.00603823 + 0.999982i $$0.501922\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ 14.7639 0.979917 0.489958 0.871746i $$-0.337012\pi$$
0.489958 + 0.871746i $$0.337012\pi$$
$$228$$ 0 0
$$229$$ 21.7082 1.43452 0.717259 0.696806i $$-0.245396\pi$$
0.717259 + 0.696806i $$0.245396\pi$$
$$230$$ 0 0
$$231$$ 3.23607 0.212918
$$232$$ 0 0
$$233$$ 2.94427 0.192886 0.0964428 0.995339i $$-0.469254\pi$$
0.0964428 + 0.995339i $$0.469254\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 8.09017 0.525513
$$238$$ 0 0
$$239$$ 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 $$0.474055\pi$$
$$242$$ 0 0
$$243$$ 16.0000 1.02640
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 1.58359 0.100762
$$248$$ 0 0
$$249$$ −6.23607 −0.395195
$$250$$ 0 0
$$251$$ 29.1803 1.84185 0.920923 0.389744i $$-0.127436\pi$$
0.920923 + 0.389744i $$0.127436\pi$$
$$252$$ 0 0
$$253$$ 19.7082 1.23904
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ −22.8541 −1.42560 −0.712800 0.701367i $$-0.752573\pi$$
−0.712800 + 0.701367i $$0.752573\pi$$
$$258$$ 0 0
$$259$$ 0.145898 0.00906566
$$260$$ 0 0
$$261$$ 7.23607 0.447901
$$262$$ 0 0
$$263$$ −10.9098 −0.672729 −0.336364 0.941732i $$-0.609197\pi$$
−0.336364 + 0.941732i $$0.609197\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −8.94427 −0.547381
$$268$$ 0 0
$$269$$ −12.7639 −0.778231 −0.389115 0.921189i $$-0.627219\pi$$
−0.389115 + 0.921189i $$0.627219\pi$$
$$270$$ 0 0
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ 0 0
$$273$$ −1.14590 −0.0693529
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 24.7082 1.48457 0.742286 0.670083i $$-0.233742\pi$$
0.742286 + 0.670083i $$0.233742\pi$$
$$278$$ 0 0
$$279$$ −6.00000 −0.359211
$$280$$ 0 0
$$281$$ 10.0902 0.601929 0.300965 0.953635i $$-0.402691\pi$$
0.300965 + 0.953635i $$0.402691\pi$$
$$282$$ 0 0
$$283$$ −29.8541 −1.77464 −0.887321 0.461152i $$-0.847436\pi$$
−0.887321 + 0.461152i $$0.847436\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −0.472136 −0.0278693
$$288$$ 0 0
$$289$$ 10.4164 0.612730
$$290$$ 0 0
$$291$$ 3.85410 0.225931
$$292$$ 0 0
$$293$$ 19.5279 1.14083 0.570415 0.821357i $$-0.306782\pi$$
0.570415 + 0.821357i $$0.306782\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ −26.1803 −1.51914
$$298$$ 0 0
$$299$$ −6.97871 −0.403589
$$300$$ 0 0
$$301$$ −3.00000 −0.172917
$$302$$ 0 0
$$303$$ 1.47214 0.0845720
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 9.23607 0.527130 0.263565 0.964642i $$-0.415102\pi$$
0.263565 + 0.964642i $$0.415102\pi$$
$$308$$ 0 0
$$309$$ 8.56231 0.487093
$$310$$ 0 0
$$311$$ −8.50658 −0.482364 −0.241182 0.970480i $$-0.577535\pi$$
−0.241182 + 0.970480i $$0.577535\pi$$
$$312$$ 0 0
$$313$$ 16.7639 0.947553 0.473777 0.880645i $$-0.342890\pi$$
0.473777 + 0.880645i $$0.342890\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −7.65248 −0.429806 −0.214903 0.976635i $$-0.568944\pi$$
−0.214903 + 0.976635i $$0.568944\pi$$
$$318$$ 0 0
$$319$$ −18.9443 −1.06068
$$320$$ 0 0
$$321$$ −16.4164 −0.916275
$$322$$ 0 0
$$323$$ −4.47214 −0.248836
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ 10.0000 0.553001
$$328$$ 0 0
$$329$$ 0.381966 0.0210585
$$330$$ 0 0
$$331$$ 23.1246 1.27104 0.635522 0.772083i $$-0.280785\pi$$
0.635522 + 0.772083i $$0.280785\pi$$
$$332$$ 0 0
$$333$$ −0.472136 −0.0258729
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −7.85410 −0.427840 −0.213920 0.976851i $$-0.568623\pi$$
−0.213920 + 0.976851i $$0.568623\pi$$
$$338$$ 0 0
$$339$$ −16.8541 −0.915389
$$340$$ 0 0
$$341$$ 15.7082 0.850647
$$342$$ 0 0
$$343$$ −8.41641 −0.454443
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ −19.9098 −1.06882 −0.534408 0.845227i $$-0.679465\pi$$
−0.534408 + 0.845227i $$0.679465\pi$$
$$348$$ 0 0
$$349$$ 21.7082 1.16201 0.581007 0.813899i $$-0.302659\pi$$
0.581007 + 0.813899i $$0.302659\pi$$
$$350$$ 0 0
$$351$$ 9.27051 0.494823
$$352$$ 0 0
$$353$$ −12.9098 −0.687121 −0.343560 0.939131i $$-0.611633\pi$$
−0.343560 + 0.939131i $$0.611633\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 3.23607 0.171271
$$358$$ 0 0
$$359$$ −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 0
$$363$$ 16.4164 0.861638
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −25.5623 −1.33434 −0.667171 0.744905i $$-0.732495\pi$$
−0.667171 + 0.744905i $$0.732495\pi$$
$$368$$ 0 0
$$369$$ 1.52786 0.0795374
$$370$$ 0 0
$$371$$ 2.14590 0.111409
$$372$$ 0 0
$$373$$ 28.2705 1.46379 0.731896 0.681417i $$-0.238636\pi$$
0.731896 + 0.681417i $$0.238636\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 6.70820 0.345490
$$378$$ 0 0
$$379$$ −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
$$382$$ 0 0
$$383$$ 33.3607 1.70465 0.852326 0.523012i $$-0.175192\pi$$
0.852326 + 0.523012i $$0.175192\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 9.70820 0.493496
$$388$$ 0 0
$$389$$ 15.0000 0.760530 0.380265 0.924878i $$-0.375833\pi$$
0.380265 + 0.924878i $$0.375833\pi$$
$$390$$ 0 0
$$391$$ 19.7082 0.996687
$$392$$ 0 0
$$393$$ −6.79837 −0.342933
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −29.0344 −1.45720 −0.728598 0.684941i $$-0.759828\pi$$
−0.728598 + 0.684941i $$0.759828\pi$$
$$398$$ 0 0
$$399$$ −0.527864 −0.0264263
$$400$$ 0 0
$$401$$ 26.5967 1.32818 0.664089 0.747653i $$-0.268820\pi$$
0.664089 + 0.747653i $$0.268820\pi$$
$$402$$ 0 0
$$403$$ −5.56231 −0.277078
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 1.23607 0.0612696
$$408$$ 0 0
$$409$$ 1.58359 0.0783036 0.0391518 0.999233i $$-0.487534\pi$$
0.0391518 + 0.999233i $$0.487534\pi$$
$$410$$ 0 0
$$411$$ 11.9443 0.589167
$$412$$ 0 0
$$413$$ 6.70820 0.330089
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ −5.00000 −0.244851
$$418$$ 0 0
$$419$$ 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 $$0.215325\pi$$
$$422$$ 0 0
$$423$$ −1.23607 −0.0600997
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 5.38197 0.260452
$$428$$ 0 0
$$429$$ −9.70820 −0.468717
$$430$$ 0 0
$$431$$ 29.8328 1.43700 0.718498 0.695529i $$-0.244830\pi$$
0.718498 + 0.695529i $$0.244830\pi$$
$$432$$ 0 0
$$433$$ −26.8541 −1.29053 −0.645263 0.763961i $$-0.723252\pi$$
−0.645263 + 0.763961i $$0.723252\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −3.21478 −0.153784
$$438$$ 0 0
$$439$$ 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
$$442$$ 0 0
$$443$$ 29.9443 1.42270 0.711348 0.702840i $$-0.248085\pi$$
0.711348 + 0.702840i $$0.248085\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −3.94427 −0.186558
$$448$$ 0 0
$$449$$ 4.67376 0.220568 0.110284 0.993900i $$-0.464824\pi$$
0.110284 + 0.993900i $$0.464824\pi$$
$$450$$ 0 0
$$451$$ −4.00000 −0.188353
$$452$$ 0 0
$$453$$ −14.5623 −0.684197
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 21.4164 1.00182 0.500909 0.865500i $$-0.332999\pi$$
0.500909 + 0.865500i $$0.332999\pi$$
$$458$$ 0 0
$$459$$ −26.1803 −1.22199
$$460$$ 0 0
$$461$$ 0.819660 0.0381754 0.0190877 0.999818i $$-0.493924\pi$$
0.0190877 + 0.999818i $$0.493924\pi$$
$$462$$ 0 0
$$463$$ −24.1246 −1.12117 −0.560583 0.828098i $$-0.689423\pi$$
−0.560583 + 0.828098i $$0.689423\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 27.4508 1.27027 0.635137 0.772400i $$-0.280944\pi$$
0.635137 + 0.772400i $$0.280944\pi$$
$$468$$ 0 0
$$469$$ 2.94427 0.135954
$$470$$ 0 0
$$471$$ −13.1803 −0.607318
$$472$$ 0 0
$$473$$ −25.4164 −1.16865
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −6.94427 −0.317956
$$478$$ 0 0
$$479$$ 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
$$482$$ 0 0
$$483$$ 2.32624 0.105847
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −36.4164 −1.65018 −0.825092 0.564998i $$-0.808877\pi$$
−0.825092 + 0.564998i $$0.808877\pi$$
$$488$$ 0 0
$$489$$ 11.0000 0.497437
$$490$$ 0 0
$$491$$ 43.2492 1.95181 0.975905 0.218196i $$-0.0700171\pi$$
0.975905 + 0.218196i $$0.0700171\pi$$
$$492$$ 0 0
$$493$$ −18.9443 −0.853207
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 4.09017 0.183469
$$498$$ 0 0
$$499$$ −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
$$502$$ 0 0
$$503$$ −37.4164 −1.66832 −0.834158 0.551526i $$-0.814046\pi$$
−0.834158 + 0.551526i $$0.814046\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −9.56231 −0.424677
$$508$$ 0 0
$$509$$ −20.3262 −0.900945 −0.450472 0.892790i $$-0.648744\pi$$
−0.450472 + 0.892790i $$0.648744\pi$$
$$510$$ 0 0
$$511$$ 5.56231 0.246062
$$512$$ 0 0
$$513$$ 4.27051 0.188548
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 3.23607 0.142322
$$518$$ 0 0
$$519$$ −18.8885 −0.829115
$$520$$ 0 0
$$521$$ 29.3607 1.28631 0.643157 0.765734i $$-0.277624\pi$$
0.643157 + 0.765734i $$0.277624\pi$$
$$522$$ 0 0
$$523$$ −13.1459 −0.574830 −0.287415 0.957806i $$-0.592796\pi$$
−0.287415 + 0.957806i $$0.592796\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ 15.7082 0.684260
$$528$$ 0 0
$$529$$ −8.83282 −0.384035
$$530$$ 0 0
$$531$$ −21.7082 −0.942056
$$532$$ 0 0
$$533$$ 1.41641 0.0613514
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ 0.527864 0.0227790
$$538$$ 0 0
$$539$$ −34.6525 −1.49259
$$540$$ 0 0
$$541$$ 27.1246 1.16618 0.583089 0.812408i $$-0.301844\pi$$
0.583089 + 0.812408i $$0.301844\pi$$
$$542$$ 0 0
$$543$$ 0.291796 0.0125222
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ −21.2918 −0.910371 −0.455186 0.890397i $$-0.650427\pi$$
−0.455186 + 0.890397i $$0.650427\pi$$
$$548$$ 0 0
$$549$$ −17.4164 −0.743314
$$550$$ 0 0
$$551$$ 3.09017 0.131646
$$552$$ 0 0
$$553$$ 5.00000 0.212622
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ −4.76393 −0.201854 −0.100927 0.994894i $$-0.532181\pi$$
−0.100927 + 0.994894i $$0.532181\pi$$
$$558$$ 0 0
$$559$$ 9.00000 0.380659
$$560$$ 0 0
$$561$$ 27.4164 1.15752
$$562$$ 0 0
$$563$$ 7.38197 0.311113 0.155556 0.987827i $$-0.450283\pi$$
0.155556 + 0.987827i $$0.450283\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 0.618034 0.0259550
$$568$$ 0 0
$$569$$ 20.5279 0.860573 0.430286 0.902692i $$-0.358413\pi$$
0.430286 + 0.902692i $$0.358413\pi$$
$$570$$ 0 0
$$571$$ 8.12461 0.340004 0.170002 0.985444i $$-0.445623\pi$$
0.170002 + 0.985444i $$0.445623\pi$$
$$572$$ 0 0
$$573$$ 1.81966 0.0760174
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 33.7771 1.40616 0.703079 0.711111i $$-0.251808\pi$$
0.703079 + 0.711111i $$0.251808\pi$$
$$578$$ 0 0
$$579$$ −7.70820 −0.320342
$$580$$ 0 0
$$581$$ −3.85410 −0.159895
$$582$$ 0 0
$$583$$ 18.1803 0.752953
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ 5.29180 0.218416 0.109208 0.994019i $$-0.465169\pi$$
0.109208 + 0.994019i $$0.465169\pi$$
$$588$$ 0 0
$$589$$ −2.56231 −0.105578
$$590$$ 0 0
$$591$$ −3.70820 −0.152535
$$592$$ 0 0
$$593$$ 10.9098 0.448013 0.224007 0.974588i $$-0.428086\pi$$
0.224007 + 0.974588i $$0.428086\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 17.5623 0.718777
$$598$$ 0 0
$$599$$ 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 $$0.482271\pi$$
$$602$$ 0 0
$$603$$ −9.52786 −0.388005
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −35.5623 −1.44343 −0.721715 0.692191i $$-0.756646\pi$$
−0.721715 + 0.692191i $$0.756646\pi$$
$$608$$ 0 0
$$609$$ −2.23607 −0.0906100
$$610$$ 0 0
$$611$$ −1.14590 −0.0463581
$$612$$ 0 0
$$613$$ 14.9787 0.604985 0.302492 0.953152i $$-0.402181\pi$$
0.302492 + 0.953152i $$0.402181\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ −14.2361 −0.573123 −0.286561 0.958062i $$-0.592512\pi$$
−0.286561 + 0.958062i $$0.592512\pi$$
$$618$$ 0 0
$$619$$ −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
$$622$$ 0 0
$$623$$ −5.52786 −0.221469
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −4.47214 −0.178600
$$628$$ 0 0
$$629$$ 1.23607 0.0492853
$$630$$ 0 0
$$631$$ 10.2361 0.407491 0.203746 0.979024i $$-0.434688\pi$$
0.203746 + 0.979024i $$0.434688\pi$$
$$632$$ 0 0
$$633$$ 9.18034 0.364886
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 12.2705 0.486175
$$638$$ 0 0
$$639$$ −13.2361 −0.523611
$$640$$ 0 0
$$641$$ −1.09017 −0.0430591 −0.0215296 0.999768i $$-0.506854\pi$$
−0.0215296 + 0.999768i $$0.506854\pi$$
$$642$$ 0 0
$$643$$ −30.8328 −1.21593 −0.607964 0.793965i $$-0.708013\pi$$
−0.607964 + 0.793965i $$0.708013\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −36.5410 −1.43658 −0.718288 0.695746i $$-0.755074\pi$$
−0.718288 + 0.695746i $$0.755074\pi$$
$$648$$ 0 0
$$649$$ 56.8328 2.23088
$$650$$ 0 0
$$651$$ 1.85410 0.0726680
$$652$$ 0 0
$$653$$ −19.0902 −0.747056 −0.373528 0.927619i $$-0.621852\pi$$
−0.373528 + 0.927619i $$0.621852\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ −18.0000 −0.702247
$$658$$ 0 0
$$659$$ −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 $$0.374937\pi$$
$$662$$ 0 0
$$663$$ −9.70820 −0.377035
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ −13.6180 −0.527292
$$668$$ 0 0
$$669$$ −0.180340 −0.00697234
$$670$$ 0 0
$$671$$ 45.5967 1.76024
$$672$$ 0 0
$$673$$ −12.1803 −0.469518 −0.234759 0.972054i $$-0.575430\pi$$
−0.234759 + 0.972054i $$0.575430\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ −10.6180 −0.408084 −0.204042 0.978962i $$-0.565408\pi$$
−0.204042 + 0.978962i $$0.565408\pi$$
$$678$$ 0 0
$$679$$ 2.38197 0.0914115
$$680$$ 0 0
$$681$$ 14.7639 0.565755
$$682$$ 0 0
$$683$$ −13.4721 −0.515497 −0.257748 0.966212i $$-0.582981\pi$$
−0.257748 + 0.966212i $$0.582981\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ 21.7082 0.828220
$$688$$ 0 0
$$689$$ −6.43769 −0.245257
$$690$$ 0 0
$$691$$ −36.2705 −1.37980 −0.689898 0.723907i $$-0.742344\pi$$
−0.689898 + 0.723907i $$0.742344\pi$$
$$692$$ 0 0
$$693$$ −6.47214 −0.245856
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −4.00000 −0.151511
$$698$$ 0 0
$$699$$ 2.94427 0.111363
$$700$$ 0 0
$$701$$ −41.0132 −1.54905 −0.774523 0.632546i $$-0.782010\pi$$
−0.774523 + 0.632546i $$0.782010\pi$$
$$702$$ 0 0
$$703$$ −0.201626 −0.00760447
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ 0.909830 0.0342177
$$708$$ 0 0
$$709$$ −33.5410 −1.25966 −0.629830 0.776733i $$-0.716875\pi$$
−0.629830 + 0.776733i $$0.716875\pi$$
$$710$$ 0 0
$$711$$ −16.1803 −0.606810
$$712$$ 0 0
$$713$$ 11.2918 0.422881
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ 20.5279 0.766627
$$718$$ 0 0
$$719$$ 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
$$722$$ 0 0
$$723$$ 2.52786 0.0940123
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 24.5623 0.910966 0.455483 0.890245i $$-0.349467\pi$$
0.455483 + 0.890245i $$0.349467\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ −25.4164 −0.940060
$$732$$ 0 0
$$733$$ 19.9787 0.737931 0.368965 0.929443i $$-0.379712\pi$$
0.368965 + 0.929443i $$0.379712\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ 24.9443 0.918834
$$738$$ 0 0
$$739$$ −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
$$742$$ 0 0
$$743$$ 28.3607 1.04045 0.520226 0.854029i $$-0.325848\pi$$
0.520226 + 0.854029i $$0.325848\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ 12.4721 0.456332
$$748$$ 0 0
$$749$$ −10.1459 −0.370723
$$750$$ 0 0
$$751$$ 5.11146 0.186520 0.0932598 0.995642i $$-0.470271\pi$$
0.0932598 + 0.995642i $$0.470271\pi$$
$$752$$ 0 0
$$753$$ 29.1803 1.06339
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −30.4164 −1.10550 −0.552752 0.833346i $$-0.686422\pi$$
−0.552752 + 0.833346i $$0.686422\pi$$
$$758$$ 0 0
$$759$$ 19.7082 0.715362
$$760$$ 0 0
$$761$$ −18.4508 −0.668843 −0.334421 0.942424i $$-0.608541\pi$$
−0.334421 + 0.942424i $$0.608541\pi$$
$$762$$ 0 0
$$763$$ 6.18034 0.223743
$$764$$ 0 0
$$765$$ 0 0
$$766$$ 0 0
$$767$$ −20.1246 −0.726658
$$768$$ 0 0
$$769$$ 13.4164 0.483808 0.241904 0.970300i $$-0.422228\pi$$
0.241904 + 0.970300i $$0.422228\pi$$
$$770$$ 0 0
$$771$$ −22.8541 −0.823070
$$772$$ 0 0
$$773$$ 36.1591 1.30055 0.650275 0.759699i $$-0.274654\pi$$
0.650275 + 0.759699i $$0.274654\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ 0.145898 0.00523406
$$778$$ 0 0
$$779$$ 0.652476 0.0233774
$$780$$ 0 0
$$781$$ 34.6525 1.23996
$$782$$ 0 0
$$783$$ 18.0902 0.646490
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −11.8197 −0.421325 −0.210663 0.977559i $$-0.567562\pi$$
−0.210663 + 0.977559i $$0.567562\pi$$
$$788$$ 0 0
$$789$$ −10.9098 −0.388400
$$790$$ 0 0
$$791$$ −10.4164 −0.370365
$$792$$ 0 0
$$793$$ −16.1459 −0.573358
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ −9.76393 −0.345856 −0.172928 0.984934i $$-0.555323\pi$$
−0.172928 + 0.984934i $$0.555323\pi$$
$$798$$ 0 0
$$799$$ 3.23607 0.114484
$$800$$ 0 0
$$801$$ 17.8885 0.632061
$$802$$ 0 0
$$803$$ 47.1246 1.66299
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ −12.7639 −0.449312
$$808$$ 0 0
$$809$$ 30.9787 1.08915 0.544577 0.838711i $$-0.316690\pi$$
0.544577 + 0.838711i $$0.316690\pi$$
$$810$$ 0 0
$$811$$ 14.7082 0.516475 0.258237 0.966081i $$-0.416858\pi$$
0.258237 + 0.966081i $$0.416858\pi$$
$$812$$ 0 0
$$813$$ 8.00000 0.280572
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 4.14590 0.145047
$$818$$ 0 0
$$819$$ 2.29180 0.0800818
$$820$$ 0 0
$$821$$ −40.6869 −1.41998 −0.709992 0.704210i $$-0.751301\pi$$
−0.709992 + 0.704210i $$0.751301\pi$$
$$822$$ 0 0
$$823$$ 47.7082 1.66300 0.831502 0.555522i $$-0.187482\pi$$
0.831502 + 0.555522i $$0.187482\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ −0.965558 −0.0335757 −0.0167879 0.999859i $$-0.505344\pi$$
−0.0167879 + 0.999859i $$0.505344\pi$$
$$828$$ 0 0
$$829$$ −35.8541 −1.24526 −0.622632 0.782515i $$-0.713937\pi$$
−0.622632 + 0.782515i $$0.713937\pi$$
$$830$$ 0 0
$$831$$ 24.7082 0.857118
$$832$$ 0 0
$$833$$ −34.6525 −1.20064
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ −15.0000 −0.518476
$$838$$ 0 0
$$839$$ −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
$$842$$ 0 0
$$843$$ 10.0902 0.347524
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 10.1459 0.348617
$$848$$ 0 0
$$849$$ −29.8541 −1.02459
$$850$$ 0 0
$$851$$ 0.888544 0.0304589
$$852$$ 0 0
$$853$$ 15.3050 0.524032 0.262016 0.965064i $$-0.415613\pi$$
0.262016 + 0.965064i $$0.415613\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −19.6869 −0.672492 −0.336246 0.941774i $$-0.609157\pi$$
−0.336246 + 0.941774i $$0.609157\pi$$
$$858$$ 0 0
$$859$$ 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
$$862$$ 0 0
$$863$$ −21.4377 −0.729748 −0.364874 0.931057i $$-0.618888\pi$$
−0.364874 + 0.931057i $$0.618888\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 10.4164 0.353760
$$868$$ 0 0
$$869$$ 42.3607 1.43699
$$870$$ 0 0
$$871$$ −8.83282 −0.299289
$$872$$ 0 0
$$873$$ −7.70820 −0.260883
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 36.5410 1.23390 0.616951 0.787001i $$-0.288368\pi$$
0.616951 + 0.787001i $$0.288368\pi$$
$$878$$ 0 0
$$879$$ 19.5279 0.658659
$$880$$ 0 0
$$881$$ −40.3607 −1.35979 −0.679893 0.733311i $$-0.737974\pi$$
−0.679893 + 0.733311i $$0.737974\pi$$
$$882$$ 0 0
$$883$$ −20.5836 −0.692693 −0.346347 0.938107i $$-0.612578\pi$$
−0.346347 + 0.938107i $$0.612578\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 29.8885 1.00356 0.501780 0.864996i $$-0.332679\pi$$
0.501780 + 0.864996i $$0.332679\pi$$
$$888$$ 0 0
$$889$$ 12.2918 0.412254
$$890$$ 0 0
$$891$$ 5.23607 0.175415
$$892$$ 0 0
$$893$$ −0.527864 −0.0176643
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −6.97871 −0.233012
$$898$$ 0 0
$$899$$ −10.8541 −0.362005
$$900$$ 0 0
$$901$$ 18.1803 0.605675
$$902$$ 0 0
$$903$$ −3.00000 −0.0998337
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −33.2492 −1.10402 −0.552011 0.833837i $$-0.686139\pi$$
−0.552011 + 0.833837i $$0.686139\pi$$
$$908$$ 0 0
$$909$$ −2.94427 −0.0976553
$$910$$ 0 0
$$911$$ 40.2361 1.33308 0.666540 0.745469i $$-0.267774\pi$$
0.666540 + 0.745469i $$0.267774\pi$$
$$912$$ 0 0
$$913$$ −32.6525 −1.08064
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ −4.20163 −0.138750
$$918$$ 0 0
$$919$$ 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
$$922$$ 0 0
$$923$$ −12.2705 −0.403889
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −17.1246 −0.562446
$$928$$ 0 0
$$929$$ 41.6312 1.36588 0.682938 0.730477i $$-0.260702\pi$$
0.682938 + 0.730477i $$0.260702\pi$$
$$930$$ 0 0
$$931$$ 5.65248 0.185252
$$932$$ 0 0
$$933$$ −8.50658 −0.278493
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ −17.7295 −0.579197 −0.289599 0.957148i $$-0.593522\pi$$
−0.289599 + 0.957148i $$0.593522\pi$$
$$938$$ 0 0
$$939$$ 16.7639 0.547070
$$940$$ 0 0
$$941$$ −46.4164 −1.51313 −0.756566 0.653918i $$-0.773124\pi$$
−0.756566 + 0.653918i $$0.773124\pi$$
$$942$$ 0 0
$$943$$ −2.87539 −0.0936355
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 2.65248 0.0861939 0.0430969 0.999071i $$-0.486278\pi$$
0.0430969 + 0.999071i $$0.486278\pi$$
$$948$$ 0 0
$$949$$ −16.6869 −0.541680
$$950$$ 0 0
$$951$$ −7.65248 −0.248149
$$952$$ 0 0
$$953$$ 7.74265 0.250809 0.125404 0.992106i $$-0.459977\pi$$
0.125404 + 0.992106i $$0.459977\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ −18.9443 −0.612381
$$958$$ 0 0
$$959$$ 7.38197 0.238376
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ 0 0
$$963$$ 32.8328 1.05802
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 4.11146 0.132216 0.0661078 0.997812i $$-0.478942\pi$$
0.0661078 + 0.997812i $$0.478942\pi$$
$$968$$ 0 0
$$969$$ −4.47214 −0.143666
$$970$$ 0 0
$$971$$ −5.61803 −0.180291 −0.0901456 0.995929i $$-0.528733\pi$$
−0.0901456 + 0.995929i $$0.528733\pi$$
$$972$$ 0 0
$$973$$ −3.09017 −0.0990663
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ 2.34752 0.0751040 0.0375520 0.999295i $$-0.488044\pi$$
0.0375520 + 0.999295i $$0.488044\pi$$
$$978$$ 0 0
$$979$$ −46.8328 −1.49678
$$980$$ 0 0
$$981$$ −20.0000 −0.638551
$$982$$ 0 0
$$983$$ 9.61803 0.306768 0.153384 0.988167i $$-0.450983\pi$$
0.153384 + 0.988167i $$0.450983\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ 0.381966 0.0121581
$$988$$ 0 0
$$989$$ −18.2705 −0.580968
$$990$$ 0 0
$$991$$ 15.3607 0.487948 0.243974 0.969782i $$-0.421549\pi$$
0.243974 + 0.969782i $$0.421549\pi$$
$$992$$ 0 0
$$993$$ 23.1246 0.733837
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −24.8885 −0.788228 −0.394114 0.919062i $$-0.628949\pi$$
−0.394114 + 0.919062i $$0.628949\pi$$
$$998$$ 0 0
$$999$$ −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 10000.2.a.l.1.2 2
4.3 odd 2 625.2.a.c.1.2 2
5.4 even 2 10000.2.a.c.1.1 2
12.11 even 2 5625.2.a.d.1.1 2
20.3 even 4 625.2.b.a.624.1 4
20.7 even 4 625.2.b.a.624.4 4
20.19 odd 2 625.2.a.b.1.1 2
25.9 even 10 400.2.u.b.81.1 4
25.14 even 10 400.2.u.b.321.1 4
60.59 even 2 5625.2.a.f.1.2 2
100.3 even 20 625.2.e.c.249.1 8
100.11 odd 10 125.2.d.a.101.1 4
100.19 odd 10 625.2.d.h.251.1 4
100.23 even 20 125.2.e.a.24.1 8
100.27 even 20 125.2.e.a.24.2 8
100.31 odd 10 625.2.d.b.251.1 4
100.39 odd 10 25.2.d.a.21.1 yes 4
100.47 even 20 625.2.e.c.249.2 8
100.59 odd 10 25.2.d.a.6.1 4
100.63 even 20 125.2.e.a.99.2 8
100.67 even 20 625.2.e.c.374.1 8
100.71 odd 10 625.2.d.b.376.1 4
100.79 odd 10 625.2.d.h.376.1 4
100.83 even 20 625.2.e.c.374.2 8
100.87 even 20 125.2.e.a.99.1 8
100.91 odd 10 125.2.d.a.26.1 4
300.59 even 10 225.2.h.b.181.1 4
300.239 even 10 225.2.h.b.46.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
25.2.d.a.6.1 4 100.59 odd 10
25.2.d.a.21.1 yes 4 100.39 odd 10
125.2.d.a.26.1 4 100.91 odd 10
125.2.d.a.101.1 4 100.11 odd 10
125.2.e.a.24.1 8 100.23 even 20
125.2.e.a.24.2 8 100.27 even 20
125.2.e.a.99.1 8 100.87 even 20
125.2.e.a.99.2 8 100.63 even 20
225.2.h.b.46.1 4 300.239 even 10
225.2.h.b.181.1 4 300.59 even 10
400.2.u.b.81.1 4 25.9 even 10
400.2.u.b.321.1 4 25.14 even 10
625.2.a.b.1.1 2 20.19 odd 2
625.2.a.c.1.2 2 4.3 odd 2
625.2.b.a.624.1 4 20.3 even 4
625.2.b.a.624.4 4 20.7 even 4
625.2.d.b.251.1 4 100.31 odd 10
625.2.d.b.376.1 4 100.71 odd 10
625.2.d.h.251.1 4 100.19 odd 10
625.2.d.h.376.1 4 100.79 odd 10
625.2.e.c.249.1 8 100.3 even 20
625.2.e.c.249.2 8 100.47 even 20
625.2.e.c.374.1 8 100.67 even 20
625.2.e.c.374.2 8 100.83 even 20
5625.2.a.d.1.1 2 12.11 even 2
5625.2.a.f.1.2 2 60.59 even 2
10000.2.a.c.1.1 2 5.4 even 2
10000.2.a.l.1.2 2 1.1 even 1 trivial