# Properties

 Label 10000.2.a.c.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: $$x^{2} - x - 1$$ 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

 $$f(q)$$ $$=$$ $$q-1.00000 q^{3} +1.61803 q^{7} -2.00000 q^{9} +O(q^{10})$$ $$q-1.00000 q^{3} +1.61803 q^{7} -2.00000 q^{9} +0.763932 q^{11} -4.85410 q^{13} -0.763932 q^{17} +5.85410 q^{19} -1.61803 q^{21} -8.23607 q^{23} +5.00000 q^{27} -1.38197 q^{29} +3.00000 q^{31} -0.763932 q^{33} +4.23607 q^{37} +4.85410 q^{39} -5.23607 q^{41} -1.85410 q^{43} +1.61803 q^{47} -4.38197 q^{49} +0.763932 q^{51} +5.47214 q^{53} -5.85410 q^{57} +4.14590 q^{59} -4.70820 q^{61} -3.23607 q^{63} -9.23607 q^{67} +8.23607 q^{69} +4.38197 q^{71} -9.00000 q^{73} +1.23607 q^{77} -3.09017 q^{79} +1.00000 q^{81} +1.76393 q^{83} +1.38197 q^{87} +8.94427 q^{89} -7.85410 q^{91} -3.00000 q^{93} +2.85410 q^{97} -1.52786 q^{99} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - 2 q^{3} + q^{7} - 4 q^{9} + O(q^{10})$$ $$2 q - 2 q^{3} + q^{7} - 4 q^{9} + 6 q^{11} - 3 q^{13} - 6 q^{17} + 5 q^{19} - q^{21} - 12 q^{23} + 10 q^{27} - 5 q^{29} + 6 q^{31} - 6 q^{33} + 4 q^{37} + 3 q^{39} - 6 q^{41} + 3 q^{43} + q^{47} - 11 q^{49} + 6 q^{51} + 2 q^{53} - 5 q^{57} + 15 q^{59} + 4 q^{61} - 2 q^{63} - 14 q^{67} + 12 q^{69} + 11 q^{71} - 18 q^{73} - 2 q^{77} + 5 q^{79} + 2 q^{81} + 8 q^{83} + 5 q^{87} - 9 q^{91} - 6 q^{93} - q^{97} - 12 q^{99} + O(q^{100})$$

## 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 $$-0.593215\pi$$
−0.288675 + 0.957427i $$0.593215\pi$$
$$4$$ 0 0
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 1.61803 0.611559 0.305780 0.952102i $$-0.401083\pi$$
0.305780 + 0.952102i $$0.401083\pi$$
$$8$$ 0 0
$$9$$ −2.00000 −0.666667
$$10$$ 0 0
$$11$$ 0.763932 0.230334 0.115167 0.993346i $$-0.463260\pi$$
0.115167 + 0.993346i $$0.463260\pi$$
$$12$$ 0 0
$$13$$ −4.85410 −1.34629 −0.673143 0.739512i $$-0.735056\pi$$
−0.673143 + 0.739512i $$0.735056\pi$$
$$14$$ 0 0
$$15$$ 0 0
$$16$$ 0 0
$$17$$ −0.763932 −0.185281 −0.0926404 0.995700i $$-0.529531\pi$$
−0.0926404 + 0.995700i $$0.529531\pi$$
$$18$$ 0 0
$$19$$ 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 0
$$23$$ −8.23607 −1.71734 −0.858669 0.512530i $$-0.828708\pi$$
−0.858669 + 0.512530i $$0.828708\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 0 0
$$27$$ 5.00000 0.962250
$$28$$ 0 0
$$29$$ −1.38197 −0.256625 −0.128312 0.991734i $$-0.540956\pi$$
−0.128312 + 0.991734i $$0.540956\pi$$
$$30$$ 0 0
$$31$$ 3.00000 0.538816 0.269408 0.963026i $$-0.413172\pi$$
0.269408 + 0.963026i $$0.413172\pi$$
$$32$$ 0 0
$$33$$ −0.763932 −0.132983
$$34$$ 0 0
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 4.23607 0.696405 0.348203 0.937419i $$-0.386792\pi$$
0.348203 + 0.937419i $$0.386792\pi$$
$$38$$ 0 0
$$39$$ 4.85410 0.777278
$$40$$ 0 0
$$41$$ −5.23607 −0.817736 −0.408868 0.912593i $$-0.634076\pi$$
−0.408868 + 0.912593i $$0.634076\pi$$
$$42$$ 0 0
$$43$$ −1.85410 −0.282748 −0.141374 0.989956i $$-0.545152\pi$$
−0.141374 + 0.989956i $$0.545152\pi$$
$$44$$ 0 0
$$45$$ 0 0
$$46$$ 0 0
$$47$$ 1.61803 0.236015 0.118007 0.993013i $$-0.462349\pi$$
0.118007 + 0.993013i $$0.462349\pi$$
$$48$$ 0 0
$$49$$ −4.38197 −0.625995
$$50$$ 0 0
$$51$$ 0.763932 0.106972
$$52$$ 0 0
$$53$$ 5.47214 0.751656 0.375828 0.926690i $$-0.377358\pi$$
0.375828 + 0.926690i $$0.377358\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 0 0
$$57$$ −5.85410 −0.775395
$$58$$ 0 0
$$59$$ 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 $$0.597458\pi$$
$$62$$ 0 0
$$63$$ −3.23607 −0.407706
$$64$$ 0 0
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −9.23607 −1.12837 −0.564183 0.825650i $$-0.690809\pi$$
−0.564183 + 0.825650i $$0.690809\pi$$
$$68$$ 0 0
$$69$$ 8.23607 0.991506
$$70$$ 0 0
$$71$$ 4.38197 0.520044 0.260022 0.965603i $$-0.416270\pi$$
0.260022 + 0.965603i $$0.416270\pi$$
$$72$$ 0 0
$$73$$ −9.00000 −1.05337 −0.526685 0.850060i $$-0.676565\pi$$
−0.526685 + 0.850060i $$0.676565\pi$$
$$74$$ 0 0
$$75$$ 0 0
$$76$$ 0 0
$$77$$ 1.23607 0.140863
$$78$$ 0 0
$$79$$ −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
$$82$$ 0 0
$$83$$ 1.76393 0.193617 0.0968083 0.995303i $$-0.469137\pi$$
0.0968083 + 0.995303i $$0.469137\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 0 0
$$87$$ 1.38197 0.148162
$$88$$ 0 0
$$89$$ 8.94427 0.948091 0.474045 0.880500i $$-0.342793\pi$$
0.474045 + 0.880500i $$0.342793\pi$$
$$90$$ 0 0
$$91$$ −7.85410 −0.823334
$$92$$ 0 0
$$93$$ −3.00000 −0.311086
$$94$$ 0 0
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 2.85410 0.289790 0.144895 0.989447i $$-0.453716\pi$$
0.144895 + 0.989447i $$0.453716\pi$$
$$98$$ 0 0
$$99$$ −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 0
$$103$$ 11.5623 1.13927 0.569634 0.821899i $$-0.307085\pi$$
0.569634 + 0.821899i $$0.307085\pi$$
$$104$$ 0 0
$$105$$ 0 0
$$106$$ 0 0
$$107$$ −10.4164 −1.00699 −0.503496 0.863998i $$-0.667953\pi$$
−0.503496 + 0.863998i $$0.667953\pi$$
$$108$$ 0 0
$$109$$ 10.0000 0.957826 0.478913 0.877862i $$-0.341031\pi$$
0.478913 + 0.877862i $$0.341031\pi$$
$$110$$ 0 0
$$111$$ −4.23607 −0.402070
$$112$$ 0 0
$$113$$ 10.1459 0.954446 0.477223 0.878782i $$-0.341643\pi$$
0.477223 + 0.878782i $$0.341643\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 0 0
$$117$$ 9.70820 0.897524
$$118$$ 0 0
$$119$$ −1.23607 −0.113310
$$120$$ 0 0
$$121$$ −10.4164 −0.946946
$$122$$ 0 0
$$123$$ 5.23607 0.472120
$$124$$ 0 0
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 15.8885 1.40988 0.704940 0.709267i $$-0.250974\pi$$
0.704940 + 0.709267i $$0.250974\pi$$
$$128$$ 0 0
$$129$$ 1.85410 0.163245
$$130$$ 0 0
$$131$$ 17.7984 1.55505 0.777526 0.628851i $$-0.216475\pi$$
0.777526 + 0.628851i $$0.216475\pi$$
$$132$$ 0 0
$$133$$ 9.47214 0.821338
$$134$$ 0 0
$$135$$ 0 0
$$136$$ 0 0
$$137$$ 5.94427 0.507853 0.253927 0.967223i $$-0.418278\pi$$
0.253927 + 0.967223i $$0.418278\pi$$
$$138$$ 0 0
$$139$$ −5.00000 −0.424094 −0.212047 0.977259i $$-0.568013\pi$$
−0.212047 + 0.977259i $$0.568013\pi$$
$$140$$ 0 0
$$141$$ −1.61803 −0.136263
$$142$$ 0 0
$$143$$ −3.70820 −0.310096
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 0 0
$$147$$ 4.38197 0.361418
$$148$$ 0 0
$$149$$ 13.9443 1.14236 0.571180 0.820825i $$-0.306486\pi$$
0.571180 + 0.820825i $$0.306486\pi$$
$$150$$ 0 0
$$151$$ 5.56231 0.452654 0.226327 0.974051i $$-0.427328\pi$$
0.226327 + 0.974051i $$0.427328\pi$$
$$152$$ 0 0
$$153$$ 1.52786 0.123520
$$154$$ 0 0
$$155$$ 0 0
$$156$$ 0 0
$$157$$ −9.18034 −0.732671 −0.366335 0.930483i $$-0.619388\pi$$
−0.366335 + 0.930483i $$0.619388\pi$$
$$158$$ 0 0
$$159$$ −5.47214 −0.433969
$$160$$ 0 0
$$161$$ −13.3262 −1.05025
$$162$$ 0 0
$$163$$ −11.0000 −0.861586 −0.430793 0.902451i $$-0.641766\pi$$
−0.430793 + 0.902451i $$0.641766\pi$$
$$164$$ 0 0
$$165$$ 0 0
$$166$$ 0 0
$$167$$ 5.56231 0.430424 0.215212 0.976567i $$-0.430956\pi$$
0.215212 + 0.976567i $$0.430956\pi$$
$$168$$ 0 0
$$169$$ 10.5623 0.812485
$$170$$ 0 0
$$171$$ −11.7082 −0.895349
$$172$$ 0 0
$$173$$ −16.8885 −1.28401 −0.642006 0.766700i $$-0.721898\pi$$
−0.642006 + 0.766700i $$0.721898\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0 0
$$177$$ −4.14590 −0.311625
$$178$$ 0 0
$$179$$ 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 $$0.329845\pi$$
$$182$$ 0 0
$$183$$ 4.70820 0.348040
$$184$$ 0 0
$$185$$ 0 0
$$186$$ 0 0
$$187$$ −0.583592 −0.0426765
$$188$$ 0 0
$$189$$ 8.09017 0.588473
$$190$$ 0 0
$$191$$ 24.1803 1.74963 0.874814 0.484459i $$-0.160984\pi$$
0.874814 + 0.484459i $$0.160984\pi$$
$$192$$ 0 0
$$193$$ −5.70820 −0.410886 −0.205443 0.978669i $$-0.565863\pi$$
−0.205443 + 0.978669i $$0.565863\pi$$
$$194$$ 0 0
$$195$$ 0 0
$$196$$ 0 0
$$197$$ −9.70820 −0.691681 −0.345840 0.938293i $$-0.612406\pi$$
−0.345840 + 0.938293i $$0.612406\pi$$
$$198$$ 0 0
$$199$$ −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
$$202$$ 0 0
$$203$$ −2.23607 −0.156941
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 0 0
$$207$$ 16.4721 1.14489
$$208$$ 0 0
$$209$$ 4.47214 0.309344
$$210$$ 0 0
$$211$$ −13.1803 −0.907372 −0.453686 0.891162i $$-0.649891\pi$$
−0.453686 + 0.891162i $$0.649891\pi$$
$$212$$ 0 0
$$213$$ −4.38197 −0.300247
$$214$$ 0 0
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 4.85410 0.329518
$$218$$ 0 0
$$219$$ 9.00000 0.608164
$$220$$ 0 0
$$221$$ 3.70820 0.249441
$$222$$ 0 0
$$223$$ −22.1803 −1.48531 −0.742653 0.669677i $$-0.766433\pi$$
−0.742653 + 0.669677i $$0.766433\pi$$
$$224$$ 0 0
$$225$$ 0 0
$$226$$ 0 0
$$227$$ −19.2361 −1.27674 −0.638371 0.769729i $$-0.720392\pi$$
−0.638371 + 0.769729i $$0.720392\pi$$
$$228$$ 0 0
$$229$$ 8.29180 0.547937 0.273969 0.961739i $$-0.411664\pi$$
0.273969 + 0.961739i $$0.411664\pi$$
$$230$$ 0 0
$$231$$ −1.23607 −0.0813273
$$232$$ 0 0
$$233$$ 14.9443 0.979032 0.489516 0.871994i $$-0.337174\pi$$
0.489516 + 0.871994i $$0.337174\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 0 0
$$237$$ 3.09017 0.200728
$$238$$ 0 0
$$239$$ 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 $$0.379532\pi$$
$$242$$ 0 0
$$243$$ −16.0000 −1.02640
$$244$$ 0 0
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −28.4164 −1.80809
$$248$$ 0 0
$$249$$ −1.76393 −0.111785
$$250$$ 0 0
$$251$$ 6.81966 0.430453 0.215227 0.976564i $$-0.430951\pi$$
0.215227 + 0.976564i $$0.430951\pi$$
$$252$$ 0 0
$$253$$ −6.29180 −0.395562
$$254$$ 0 0
$$255$$ 0 0
$$256$$ 0 0
$$257$$ 16.1459 1.00715 0.503577 0.863951i $$-0.332017\pi$$
0.503577 + 0.863951i $$0.332017\pi$$
$$258$$ 0 0
$$259$$ 6.85410 0.425893
$$260$$ 0 0
$$261$$ 2.76393 0.171083
$$262$$ 0 0
$$263$$ 22.0902 1.36214 0.681069 0.732219i $$-0.261515\pi$$
0.681069 + 0.732219i $$0.261515\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 0 0
$$267$$ −8.94427 −0.547381
$$268$$ 0 0
$$269$$ −17.2361 −1.05090 −0.525451 0.850824i $$-0.676103\pi$$
−0.525451 + 0.850824i $$0.676103\pi$$
$$270$$ 0 0
$$271$$ 8.00000 0.485965 0.242983 0.970031i $$-0.421874\pi$$
0.242983 + 0.970031i $$0.421874\pi$$
$$272$$ 0 0
$$273$$ 7.85410 0.475352
$$274$$ 0 0
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −11.2918 −0.678458 −0.339229 0.940704i $$-0.610166\pi$$
−0.339229 + 0.940704i $$0.610166\pi$$
$$278$$ 0 0
$$279$$ −6.00000 −0.359211
$$280$$ 0 0
$$281$$ −1.09017 −0.0650341 −0.0325170 0.999471i $$-0.510352\pi$$
−0.0325170 + 0.999471i $$0.510352\pi$$
$$282$$ 0 0
$$283$$ 23.1459 1.37588 0.687940 0.725767i $$-0.258515\pi$$
0.687940 + 0.725767i $$0.258515\pi$$
$$284$$ 0 0
$$285$$ 0 0
$$286$$ 0 0
$$287$$ −8.47214 −0.500094
$$288$$ 0 0
$$289$$ −16.4164 −0.965671
$$290$$ 0 0
$$291$$ −2.85410 −0.167310
$$292$$ 0 0
$$293$$ −28.4721 −1.66336 −0.831680 0.555255i $$-0.812621\pi$$
−0.831680 + 0.555255i $$0.812621\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 0 0
$$297$$ 3.81966 0.221639
$$298$$ 0 0
$$299$$ 39.9787 2.31203
$$300$$ 0 0
$$301$$ −3.00000 −0.172917
$$302$$ 0 0
$$303$$ 7.47214 0.429263
$$304$$ 0 0
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −4.76393 −0.271892 −0.135946 0.990716i $$-0.543407\pi$$
−0.135946 + 0.990716i $$0.543407\pi$$
$$308$$ 0 0
$$309$$ −11.5623 −0.657757
$$310$$ 0 0
$$311$$ 29.5066 1.67316 0.836582 0.547841i $$-0.184550\pi$$
0.836582 + 0.547841i $$0.184550\pi$$
$$312$$ 0 0
$$313$$ −21.2361 −1.20033 −0.600167 0.799875i $$-0.704899\pi$$
−0.600167 + 0.799875i $$0.704899\pi$$
$$314$$ 0 0
$$315$$ 0 0
$$316$$ 0 0
$$317$$ −23.6525 −1.32846 −0.664228 0.747530i $$-0.731239\pi$$
−0.664228 + 0.747530i $$0.731239\pi$$
$$318$$ 0 0
$$319$$ −1.05573 −0.0591094
$$320$$ 0 0
$$321$$ 10.4164 0.581387
$$322$$ 0 0
$$323$$ −4.47214 −0.248836
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 0 0
$$327$$ −10.0000 −0.553001
$$328$$ 0 0
$$329$$ 2.61803 0.144337
$$330$$ 0 0
$$331$$ −17.1246 −0.941254 −0.470627 0.882332i $$-0.655972\pi$$
−0.470627 + 0.882332i $$0.655972\pi$$
$$332$$ 0 0
$$333$$ −8.47214 −0.464270
$$334$$ 0 0
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 1.14590 0.0624210 0.0312105 0.999513i $$-0.490064\pi$$
0.0312105 + 0.999513i $$0.490064\pi$$
$$338$$ 0 0
$$339$$ −10.1459 −0.551050
$$340$$ 0 0
$$341$$ 2.29180 0.124108
$$342$$ 0 0
$$343$$ −18.4164 −0.994393
$$344$$ 0 0
$$345$$ 0 0
$$346$$ 0 0
$$347$$ 31.0902 1.66901 0.834504 0.551002i $$-0.185754\pi$$
0.834504 + 0.551002i $$0.185754\pi$$
$$348$$ 0 0
$$349$$ 8.29180 0.443850 0.221925 0.975064i $$-0.428766\pi$$
0.221925 + 0.975064i $$0.428766\pi$$
$$350$$ 0 0
$$351$$ −24.2705 −1.29546
$$352$$ 0 0
$$353$$ 24.0902 1.28219 0.641095 0.767461i $$-0.278480\pi$$
0.641095 + 0.767461i $$0.278480\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 1.23607 0.0654197
$$358$$ 0 0
$$359$$ 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
$$362$$ 0 0
$$363$$ 10.4164 0.546720
$$364$$ 0 0
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 5.43769 0.283845 0.141923 0.989878i $$-0.454672\pi$$
0.141923 + 0.989878i $$0.454672\pi$$
$$368$$ 0 0
$$369$$ 10.4721 0.545158
$$370$$ 0 0
$$371$$ 8.85410 0.459682
$$372$$ 0 0
$$373$$ 5.27051 0.272897 0.136448 0.990647i $$-0.456431\pi$$
0.136448 + 0.990647i $$0.456431\pi$$
$$374$$ 0 0
$$375$$ 0 0
$$376$$ 0 0
$$377$$ 6.70820 0.345490
$$378$$ 0 0
$$379$$ 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
$$382$$ 0 0
$$383$$ 11.3607 0.580504 0.290252 0.956950i $$-0.406261\pi$$
0.290252 + 0.956950i $$0.406261\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 0 0
$$387$$ 3.70820 0.188499
$$388$$ 0 0
$$389$$ 15.0000 0.760530 0.380265 0.924878i $$-0.375833\pi$$
0.380265 + 0.924878i $$0.375833\pi$$
$$390$$ 0 0
$$391$$ 6.29180 0.318190
$$392$$ 0 0
$$393$$ −17.7984 −0.897809
$$394$$ 0 0
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −0.0344419 −0.00172859 −0.000864294 1.00000i $$-0.500275\pi$$
−0.000864294 1.00000i $$0.500275\pi$$
$$398$$ 0 0
$$399$$ −9.47214 −0.474200
$$400$$ 0 0
$$401$$ −22.5967 −1.12843 −0.564214 0.825629i $$-0.690821\pi$$
−0.564214 + 0.825629i $$0.690821\pi$$
$$402$$ 0 0
$$403$$ −14.5623 −0.725400
$$404$$ 0 0
$$405$$ 0 0
$$406$$ 0 0
$$407$$ 3.23607 0.160406
$$408$$ 0 0
$$409$$ 28.4164 1.40510 0.702550 0.711634i $$-0.252045\pi$$
0.702550 + 0.711634i $$0.252045\pi$$
$$410$$ 0 0
$$411$$ −5.94427 −0.293209
$$412$$ 0 0
$$413$$ 6.70820 0.330089
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 0 0
$$417$$ 5.00000 0.244851
$$418$$ 0 0
$$419$$ 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 $$0.215325\pi$$
$$422$$ 0 0
$$423$$ −3.23607 −0.157343
$$424$$ 0 0
$$425$$ 0 0
$$426$$ 0 0
$$427$$ −7.61803 −0.368663
$$428$$ 0 0
$$429$$ 3.70820 0.179034
$$430$$ 0 0
$$431$$ −23.8328 −1.14799 −0.573993 0.818860i $$-0.694606\pi$$
−0.573993 + 0.818860i $$0.694606\pi$$
$$432$$ 0 0
$$433$$ 20.1459 0.968150 0.484075 0.875026i $$-0.339156\pi$$
0.484075 + 0.875026i $$0.339156\pi$$
$$434$$ 0 0
$$435$$ 0 0
$$436$$ 0 0
$$437$$ −48.2148 −2.30643
$$438$$ 0 0
$$439$$ −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
$$442$$ 0 0
$$443$$ −12.0557 −0.572785 −0.286392 0.958112i $$-0.592456\pi$$
−0.286392 + 0.958112i $$0.592456\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 0 0
$$447$$ −13.9443 −0.659541
$$448$$ 0 0
$$449$$ 20.3262 0.959254 0.479627 0.877472i $$-0.340772\pi$$
0.479627 + 0.877472i $$0.340772\pi$$
$$450$$ 0 0
$$451$$ −4.00000 −0.188353
$$452$$ 0 0
$$453$$ −5.56231 −0.261340
$$454$$ 0 0
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 5.41641 0.253369 0.126684 0.991943i $$-0.459566\pi$$
0.126684 + 0.991943i $$0.459566\pi$$
$$458$$ 0 0
$$459$$ −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 0
$$463$$ −16.1246 −0.749374 −0.374687 0.927151i $$-0.622250\pi$$
−0.374687 + 0.927151i $$0.622250\pi$$
$$464$$ 0 0
$$465$$ 0 0
$$466$$ 0 0
$$467$$ 28.4508 1.31655 0.658274 0.752778i $$-0.271287\pi$$
0.658274 + 0.752778i $$0.271287\pi$$
$$468$$ 0 0
$$469$$ −14.9443 −0.690062
$$470$$ 0 0
$$471$$ 9.18034 0.423008
$$472$$ 0 0
$$473$$ −1.41641 −0.0651265
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 0 0
$$477$$ −10.9443 −0.501104
$$478$$ 0 0
$$479$$ 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
$$482$$ 0 0
$$483$$ 13.3262 0.606365
$$484$$ 0 0
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 9.58359 0.434274 0.217137 0.976141i $$-0.430328\pi$$
0.217137 + 0.976141i $$0.430328\pi$$
$$488$$ 0 0
$$489$$ 11.0000 0.497437
$$490$$ 0 0
$$491$$ −37.2492 −1.68103 −0.840517 0.541785i $$-0.817749\pi$$
−0.840517 + 0.541785i $$0.817749\pi$$
$$492$$ 0 0
$$493$$ 1.05573 0.0475476
$$494$$ 0 0
$$495$$ 0 0
$$496$$ 0 0
$$497$$ 7.09017 0.318038
$$498$$ 0 0
$$499$$ 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
$$502$$ 0 0
$$503$$ 10.5836 0.471899 0.235950 0.971765i $$-0.424180\pi$$
0.235950 + 0.971765i $$0.424180\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 0 0
$$507$$ −10.5623 −0.469088
$$508$$ 0 0
$$509$$ −4.67376 −0.207161 −0.103580 0.994621i $$-0.533030\pi$$
−0.103580 + 0.994621i $$0.533030\pi$$
$$510$$ 0 0
$$511$$ −14.5623 −0.644198
$$512$$ 0 0
$$513$$ 29.2705 1.29232
$$514$$ 0 0
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 1.23607 0.0543622
$$518$$ 0 0
$$519$$ 16.8885 0.741325
$$520$$ 0 0
$$521$$ −15.3607 −0.672964 −0.336482 0.941690i $$-0.609237\pi$$
−0.336482 + 0.941690i $$0.609237\pi$$
$$522$$ 0 0
$$523$$ 19.8541 0.868159 0.434080 0.900875i $$-0.357074\pi$$
0.434080 + 0.900875i $$0.357074\pi$$
$$524$$ 0 0
$$525$$ 0 0
$$526$$ 0 0
$$527$$ −2.29180 −0.0998322
$$528$$ 0 0
$$529$$ 44.8328 1.94925
$$530$$ 0 0
$$531$$ −8.29180 −0.359833
$$532$$ 0 0
$$533$$ 25.4164 1.10091
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 0 0
$$537$$ −9.47214 −0.408753
$$538$$ 0 0
$$539$$ −3.34752 −0.144188
$$540$$ 0 0
$$541$$ −13.1246 −0.564271 −0.282136 0.959375i $$-0.591043\pi$$
−0.282136 + 0.959375i $$0.591043\pi$$
$$542$$ 0 0
$$543$$ −13.7082 −0.588275
$$544$$ 0 0
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 34.7082 1.48402 0.742008 0.670391i $$-0.233874\pi$$
0.742008 + 0.670391i $$0.233874\pi$$
$$548$$ 0 0
$$549$$ 9.41641 0.401882
$$550$$ 0 0
$$551$$ −8.09017 −0.344653
$$552$$ 0 0
$$553$$ −5.00000 −0.212622
$$554$$ 0 0
$$555$$ 0 0
$$556$$ 0 0
$$557$$ 9.23607 0.391345 0.195672 0.980669i $$-0.437311\pi$$
0.195672 + 0.980669i $$0.437311\pi$$
$$558$$ 0 0
$$559$$ 9.00000 0.380659
$$560$$ 0 0
$$561$$ 0.583592 0.0246393
$$562$$ 0 0
$$563$$ −9.61803 −0.405352 −0.202676 0.979246i $$-0.564964\pi$$
−0.202676 + 0.979246i $$0.564964\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 0 0
$$567$$ 1.61803 0.0679510
$$568$$ 0 0
$$569$$ 29.4721 1.23554 0.617768 0.786360i $$-0.288037\pi$$
0.617768 + 0.786360i $$0.288037\pi$$
$$570$$ 0 0
$$571$$ −32.1246 −1.34437 −0.672187 0.740382i $$-0.734645\pi$$
−0.672187 + 0.740382i $$0.734645\pi$$
$$572$$ 0 0
$$573$$ −24.1803 −1.01015
$$574$$ 0 0
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 37.7771 1.57268 0.786340 0.617794i $$-0.211973\pi$$
0.786340 + 0.617794i $$0.211973\pi$$
$$578$$ 0 0
$$579$$ 5.70820 0.237225
$$580$$ 0 0
$$581$$ 2.85410 0.118408
$$582$$ 0 0
$$583$$ 4.18034 0.173132
$$584$$ 0 0
$$585$$ 0 0
$$586$$ 0 0
$$587$$ −18.7082 −0.772170 −0.386085 0.922463i $$-0.626173\pi$$
−0.386085 + 0.922463i $$0.626173\pi$$
$$588$$ 0 0
$$589$$ 17.5623 0.723642
$$590$$ 0 0
$$591$$ 9.70820 0.399342
$$592$$ 0 0
$$593$$ −22.0902 −0.907135 −0.453567 0.891222i $$-0.649849\pi$$
−0.453567 + 0.891222i $$0.649849\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 0 0
$$597$$ 2.56231 0.104868
$$598$$ 0 0
$$599$$ 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 $$0.234943\pi$$
$$602$$ 0 0
$$603$$ 18.4721 0.752244
$$604$$ 0 0
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 15.4377 0.626597 0.313298 0.949655i $$-0.398566\pi$$
0.313298 + 0.949655i $$0.398566\pi$$
$$608$$ 0 0
$$609$$ 2.23607 0.0906100
$$610$$ 0 0
$$611$$ −7.85410 −0.317743
$$612$$ 0 0
$$613$$ 31.9787 1.29161 0.645804 0.763503i $$-0.276522\pi$$
0.645804 + 0.763503i $$0.276522\pi$$
$$614$$ 0 0
$$615$$ 0 0
$$616$$ 0 0
$$617$$ 9.76393 0.393081 0.196541 0.980496i $$-0.437029\pi$$
0.196541 + 0.980496i $$0.437029\pi$$
$$618$$ 0 0
$$619$$ −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
$$622$$ 0 0
$$623$$ 14.4721 0.579814
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 0 0
$$627$$ −4.47214 −0.178600
$$628$$ 0 0
$$629$$ −3.23607 −0.129030
$$630$$ 0 0
$$631$$ 5.76393 0.229459 0.114729 0.993397i $$-0.463400\pi$$
0.114729 + 0.993397i $$0.463400\pi$$
$$632$$ 0 0
$$633$$ 13.1803 0.523871
$$634$$ 0 0
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 21.2705 0.842768
$$638$$ 0 0
$$639$$ −8.76393 −0.346696
$$640$$ 0 0
$$641$$ 10.0902 0.398538 0.199269 0.979945i $$-0.436143\pi$$
0.199269 + 0.979945i $$0.436143\pi$$
$$642$$ 0 0
$$643$$ −22.8328 −0.900438 −0.450219 0.892918i $$-0.648654\pi$$
−0.450219 + 0.892918i $$0.648654\pi$$
$$644$$ 0 0
$$645$$ 0 0
$$646$$ 0 0
$$647$$ −30.5410 −1.20069 −0.600346 0.799741i $$-0.704970\pi$$
−0.600346 + 0.799741i $$0.704970\pi$$
$$648$$ 0 0
$$649$$ 3.16718 0.124323
$$650$$ 0 0
$$651$$ −4.85410 −0.190247
$$652$$ 0 0
$$653$$ 7.90983 0.309536 0.154768 0.987951i $$-0.450537\pi$$
0.154768 + 0.987951i $$0.450537\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 0 0
$$657$$ 18.0000 0.702247
$$658$$ 0 0
$$659$$ −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 $$0.790579\pi$$
$$662$$ 0 0
$$663$$ −3.70820 −0.144015
$$664$$ 0 0
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 11.3820 0.440711
$$668$$ 0 0
$$669$$ 22.1803 0.857541
$$670$$ 0 0
$$671$$ −3.59675 −0.138851
$$672$$ 0 0
$$673$$ −10.1803 −0.392423 −0.196212 0.980562i $$-0.562864\pi$$
−0.196212 + 0.980562i $$0.562864\pi$$
$$674$$ 0 0
$$675$$ 0 0
$$676$$ 0 0
$$677$$ 8.38197 0.322145 0.161073 0.986943i $$-0.448505\pi$$
0.161073 + 0.986943i $$0.448505\pi$$
$$678$$ 0 0
$$679$$ 4.61803 0.177224
$$680$$ 0 0
$$681$$ 19.2361 0.737128
$$682$$ 0 0
$$683$$ 4.52786 0.173254 0.0866270 0.996241i $$-0.472391\pi$$
0.0866270 + 0.996241i $$0.472391\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 0 0
$$687$$ −8.29180 −0.316352
$$688$$ 0 0
$$689$$ −26.5623 −1.01194
$$690$$ 0 0
$$691$$ −2.72949 −0.103835 −0.0519173 0.998651i $$-0.516533\pi$$
−0.0519173 + 0.998651i $$0.516533\pi$$
$$692$$ 0 0
$$693$$ −2.47214 −0.0939087
$$694$$ 0 0
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 4.00000 0.151511
$$698$$ 0 0
$$699$$ −14.9443 −0.565244
$$700$$ 0 0
$$701$$ 35.0132 1.32243 0.661214 0.750197i $$-0.270041\pi$$
0.661214 + 0.750197i $$0.270041\pi$$
$$702$$ 0 0
$$703$$ 24.7984 0.935288
$$704$$ 0 0
$$705$$ 0 0
$$706$$ 0 0
$$707$$ −12.0902 −0.454698
$$708$$ 0 0
$$709$$ 33.5410 1.25966 0.629830 0.776733i $$-0.283125\pi$$
0.629830 + 0.776733i $$0.283125\pi$$
$$710$$ 0 0
$$711$$ 6.18034 0.231781
$$712$$ 0 0
$$713$$ −24.7082 −0.925330
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 0 0
$$717$$ −29.4721 −1.10066
$$718$$ 0 0
$$719$$ 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
$$722$$ 0 0
$$723$$ −11.4721 −0.426653
$$724$$ 0 0
$$725$$ 0 0
$$726$$ 0 0
$$727$$ −4.43769 −0.164585 −0.0822925 0.996608i $$-0.526224\pi$$
−0.0822925 + 0.996608i $$0.526224\pi$$
$$728$$ 0 0
$$729$$ 13.0000 0.481481
$$730$$ 0 0
$$731$$ 1.41641 0.0523877
$$732$$ 0 0
$$733$$ 26.9787 0.996482 0.498241 0.867039i $$-0.333980\pi$$
0.498241 + 0.867039i $$0.333980\pi$$
$$734$$ 0 0
$$735$$ 0 0
$$736$$ 0 0
$$737$$ −7.05573 −0.259901
$$738$$ 0 0
$$739$$ 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
$$742$$ 0 0
$$743$$ 16.3607 0.600215 0.300108 0.953905i $$-0.402977\pi$$
0.300108 + 0.953905i $$0.402977\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 0 0
$$747$$ −3.52786 −0.129078
$$748$$ 0 0
$$749$$ −16.8541 −0.615835
$$750$$ 0 0
$$751$$ 40.8885 1.49204 0.746022 0.665921i $$-0.231961\pi$$
0.746022 + 0.665921i $$0.231961\pi$$
$$752$$ 0 0
$$753$$ −6.81966 −0.248522
$$754$$ 0 0
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 3.58359 0.130248 0.0651239 0.997877i $$-0.479256\pi$$
0.0651239 + 0.997877i $$0.479256\pi$$
$$758$$ 0 0
$$759$$ 6.29180 0.228378
$$760$$ 0 0
$$761$$ 37.4508 1.35759 0.678796 0.734327i $$-0.262502\pi$$
0.678796 + 0.734327i $$0.262502\pi$$
$$762$$ 0 0
$$763$$ 16.1803 0.585768
$$764$$ 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 $$-0.577772\pi$$
−0.241904 + 0.970300i $$0.577772\pi$$
$$770$$ 0 0
$$771$$ −16.1459 −0.581480
$$772$$ 0 0
$$773$$ 33.1591 1.19265 0.596324 0.802744i $$-0.296627\pi$$
0.596324 + 0.802744i $$0.296627\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 0 0
$$777$$ −6.85410 −0.245890
$$778$$ 0 0
$$779$$ −30.6525 −1.09824
$$780$$ 0 0
$$781$$ 3.34752 0.119784
$$782$$ 0 0
$$783$$ −6.90983 −0.246937
$$784$$ 0 0
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 34.1803 1.21840 0.609199 0.793018i $$-0.291491\pi$$
0.609199 + 0.793018i $$0.291491\pi$$
$$788$$ 0 0
$$789$$ −22.0902 −0.786431
$$790$$ 0 0
$$791$$ 16.4164 0.583700
$$792$$ 0 0
$$793$$ 22.8541 0.811573
$$794$$ 0 0
$$795$$ 0 0
$$796$$ 0 0
$$797$$ 14.2361 0.504267 0.252134 0.967692i $$-0.418868\pi$$
0.252134 + 0.967692i $$0.418868\pi$$
$$798$$ 0 0
$$799$$ −1.23607 −0.0437289
$$800$$ 0 0
$$801$$ −17.8885 −0.632061
$$802$$ 0 0
$$803$$ −6.87539 −0.242627
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 0 0
$$807$$ 17.2361 0.606738
$$808$$ 0 0
$$809$$ −15.9787 −0.561782 −0.280891 0.959740i $$-0.590630\pi$$
−0.280891 + 0.959740i $$0.590630\pi$$
$$810$$ 0 0
$$811$$ 1.29180 0.0453611 0.0226805 0.999743i $$-0.492780\pi$$
0.0226805 + 0.999743i $$0.492780\pi$$
$$812$$ 0 0
$$813$$ −8.00000 −0.280572
$$814$$ 0 0
$$815$$ 0 0
$$816$$ 0 0
$$817$$ −10.8541 −0.379737
$$818$$ 0 0
$$819$$ 15.7082 0.548889
$$820$$ 0 0
$$821$$ 19.6869 0.687078 0.343539 0.939138i $$-0.388374\pi$$
0.343539 + 0.939138i $$0.388374\pi$$
$$822$$ 0 0
$$823$$ −34.2918 −1.19534 −0.597668 0.801743i $$-0.703906\pi$$
−0.597668 + 0.801743i $$0.703906\pi$$
$$824$$ 0 0
$$825$$ 0 0
$$826$$ 0 0
$$827$$ 30.0344 1.04440 0.522200 0.852823i $$-0.325111\pi$$
0.522200 + 0.852823i $$0.325111\pi$$
$$828$$ 0 0
$$829$$ −29.1459 −1.01228 −0.506139 0.862452i $$-0.668928\pi$$
−0.506139 + 0.862452i $$0.668928\pi$$
$$830$$ 0 0
$$831$$ 11.2918 0.391708
$$832$$ 0 0
$$833$$ 3.34752 0.115985
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 0 0
$$837$$ 15.0000 0.518476
$$838$$ 0 0
$$839$$ −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
$$842$$ 0 0
$$843$$ 1.09017 0.0375474
$$844$$ 0 0
$$845$$ 0 0
$$846$$ 0 0
$$847$$ −16.8541 −0.579114
$$848$$ 0 0
$$849$$ −23.1459 −0.794365
$$850$$ 0 0
$$851$$ −34.8885 −1.19596
$$852$$ 0 0
$$853$$ 47.3050 1.61969 0.809845 0.586643i $$-0.199551\pi$$
0.809845 + 0.586643i $$0.199551\pi$$
$$854$$ 0 0
$$855$$ 0 0
$$856$$ 0 0
$$857$$ −40.6869 −1.38984 −0.694919 0.719088i $$-0.744560\pi$$
−0.694919 + 0.719088i $$0.744560\pi$$
$$858$$ 0 0
$$859$$ 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
$$862$$ 0 0
$$863$$ 41.5623 1.41480 0.707399 0.706815i $$-0.249869\pi$$
0.707399 + 0.706815i $$0.249869\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 0 0
$$867$$ 16.4164 0.557530
$$868$$ 0 0
$$869$$ −2.36068 −0.0800806
$$870$$ 0 0
$$871$$ 44.8328 1.51910
$$872$$ 0 0
$$873$$ −5.70820 −0.193193
$$874$$ 0 0
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 30.5410 1.03130 0.515648 0.856800i $$-0.327551\pi$$
0.515648 + 0.856800i $$0.327551\pi$$
$$878$$ 0 0
$$879$$ 28.4721 0.960341
$$880$$ 0 0
$$881$$ 4.36068 0.146915 0.0734575 0.997298i $$-0.476597\pi$$
0.0734575 + 0.997298i $$0.476597\pi$$
$$882$$ 0 0
$$883$$ 47.4164 1.59569 0.797845 0.602863i $$-0.205974\pi$$
0.797845 + 0.602863i $$0.205974\pi$$
$$884$$ 0 0
$$885$$ 0 0
$$886$$ 0 0
$$887$$ 5.88854 0.197718 0.0988590 0.995101i $$-0.468481\pi$$
0.0988590 + 0.995101i $$0.468481\pi$$
$$888$$ 0 0
$$889$$ 25.7082 0.862225
$$890$$ 0 0
$$891$$ 0.763932 0.0255927
$$892$$ 0 0
$$893$$ 9.47214 0.316973
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 0 0
$$897$$ −39.9787 −1.33485
$$898$$ 0 0
$$899$$ −4.14590 −0.138273
$$900$$ 0 0
$$901$$ −4.18034 −0.139267
$$902$$ 0 0
$$903$$ 3.00000 0.0998337
$$904$$ 0 0
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −47.2492 −1.56888 −0.784442 0.620202i $$-0.787051\pi$$
−0.784442 + 0.620202i $$0.787051\pi$$
$$908$$ 0 0
$$909$$ 14.9443 0.495670
$$910$$ 0 0
$$911$$ 35.7639 1.18491 0.592456 0.805603i $$-0.298158\pi$$
0.592456 + 0.805603i $$0.298158\pi$$
$$912$$ 0 0
$$913$$ 1.34752 0.0445965
$$914$$ 0 0
$$915$$ 0 0
$$916$$ 0 0
$$917$$ 28.7984 0.951006
$$918$$ 0 0
$$919$$ 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
$$922$$ 0 0
$$923$$ −21.2705 −0.700127
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 0 0
$$927$$ −23.1246 −0.759512
$$928$$ 0 0
$$929$$ −36.6312 −1.20183 −0.600915 0.799313i $$-0.705197\pi$$
−0.600915 + 0.799313i $$0.705197\pi$$
$$930$$ 0 0
$$931$$ −25.6525 −0.840726
$$932$$ 0 0
$$933$$ −29.5066 −0.966002
$$934$$ 0 0
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 51.2705 1.67493 0.837467 0.546487i $$-0.184035\pi$$
0.837467 + 0.546487i $$0.184035\pi$$
$$938$$ 0 0
$$939$$ 21.2361 0.693013
$$940$$ 0 0
$$941$$ −19.5836 −0.638407 −0.319203 0.947686i $$-0.603415\pi$$
−0.319203 + 0.947686i $$0.603415\pi$$
$$942$$ 0 0
$$943$$ 43.1246 1.40433
$$944$$ 0 0
$$945$$ 0 0
$$946$$ 0 0
$$947$$ 28.6525 0.931080 0.465540 0.885027i $$-0.345860\pi$$
0.465540 + 0.885027i $$0.345860\pi$$
$$948$$ 0 0
$$949$$ 43.6869 1.41814
$$950$$ 0 0
$$951$$ 23.6525 0.766984
$$952$$ 0 0
$$953$$ 34.7426 1.12542 0.562712 0.826653i $$-0.309758\pi$$
0.562712 + 0.826653i $$0.309758\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 0 0
$$957$$ 1.05573 0.0341268
$$958$$ 0 0
$$959$$ 9.61803 0.310583
$$960$$ 0 0
$$961$$ −22.0000 −0.709677
$$962$$ 0 0
$$963$$ 20.8328 0.671328
$$964$$ 0 0
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −39.8885 −1.28273 −0.641365 0.767236i $$-0.721631\pi$$
−0.641365 + 0.767236i $$0.721631\pi$$
$$968$$ 0 0
$$969$$ 4.47214 0.143666
$$970$$ 0 0
$$971$$ −3.38197 −0.108532 −0.0542662 0.998527i $$-0.517282\pi$$
−0.0542662 + 0.998527i $$0.517282\pi$$
$$972$$ 0 0
$$973$$ −8.09017 −0.259359
$$974$$ 0 0
$$975$$ 0 0
$$976$$ 0 0
$$977$$ −33.6525 −1.07664 −0.538319 0.842741i $$-0.680940\pi$$
−0.538319 + 0.842741i $$0.680940\pi$$
$$978$$ 0 0
$$979$$ 6.83282 0.218378
$$980$$ 0 0
$$981$$ −20.0000 −0.638551
$$982$$ 0 0
$$983$$ −7.38197 −0.235448 −0.117724 0.993046i $$-0.537560\pi$$
−0.117724 + 0.993046i $$0.537560\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 0 0
$$987$$ −2.61803 −0.0833329
$$988$$ 0 0
$$989$$ 15.2705 0.485574
$$990$$ 0 0
$$991$$ −29.3607 −0.932673 −0.466336 0.884607i $$-0.654426\pi$$
−0.466336 + 0.884607i $$0.654426\pi$$
$$992$$ 0 0
$$993$$ 17.1246 0.543433
$$994$$ 0 0
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −10.8885 −0.344844 −0.172422 0.985023i $$-0.555159\pi$$
−0.172422 + 0.985023i $$0.555159\pi$$
$$998$$ 0 0
$$999$$ 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 10000.2.a.c.1.2 2
4.3 odd 2 625.2.a.b.1.2 2
5.4 even 2 10000.2.a.l.1.1 2
12.11 even 2 5625.2.a.f.1.1 2
20.3 even 4 625.2.b.a.624.2 4
20.7 even 4 625.2.b.a.624.3 4
20.19 odd 2 625.2.a.c.1.1 2
25.6 even 5 400.2.u.b.161.1 4
25.21 even 5 400.2.u.b.241.1 4
60.59 even 2 5625.2.a.d.1.2 2
100.3 even 20 125.2.e.a.49.1 8
100.11 odd 10 625.2.d.h.501.1 4
100.19 odd 10 125.2.d.a.51.1 4
100.23 even 20 625.2.e.c.124.1 8
100.27 even 20 625.2.e.c.124.2 8
100.31 odd 10 25.2.d.a.11.1 4
100.39 odd 10 625.2.d.b.501.1 4
100.47 even 20 125.2.e.a.49.2 8
100.59 odd 10 625.2.d.b.126.1 4
100.63 even 20 625.2.e.c.499.2 8
100.67 even 20 125.2.e.a.74.1 8
100.71 odd 10 25.2.d.a.16.1 yes 4
100.79 odd 10 125.2.d.a.76.1 4
100.83 even 20 125.2.e.a.74.2 8
100.87 even 20 625.2.e.c.499.1 8
100.91 odd 10 625.2.d.h.126.1 4
300.71 even 10 225.2.h.b.91.1 4
300.131 even 10 225.2.h.b.136.1 4

By twisted newform
Twist Min Dim Char Parity Ord Type
25.2.d.a.11.1 4 100.31 odd 10
25.2.d.a.16.1 yes 4 100.71 odd 10
125.2.d.a.51.1 4 100.19 odd 10
125.2.d.a.76.1 4 100.79 odd 10
125.2.e.a.49.1 8 100.3 even 20
125.2.e.a.49.2 8 100.47 even 20
125.2.e.a.74.1 8 100.67 even 20
125.2.e.a.74.2 8 100.83 even 20
225.2.h.b.91.1 4 300.71 even 10
225.2.h.b.136.1 4 300.131 even 10
400.2.u.b.161.1 4 25.6 even 5
400.2.u.b.241.1 4 25.21 even 5
625.2.a.b.1.2 2 4.3 odd 2
625.2.a.c.1.1 2 20.19 odd 2
625.2.b.a.624.2 4 20.3 even 4
625.2.b.a.624.3 4 20.7 even 4
625.2.d.b.126.1 4 100.59 odd 10
625.2.d.b.501.1 4 100.39 odd 10
625.2.d.h.126.1 4 100.91 odd 10
625.2.d.h.501.1 4 100.11 odd 10
625.2.e.c.124.1 8 100.23 even 20
625.2.e.c.124.2 8 100.27 even 20
625.2.e.c.499.1 8 100.87 even 20
625.2.e.c.499.2 8 100.63 even 20
5625.2.a.d.1.2 2 60.59 even 2
5625.2.a.f.1.1 2 12.11 even 2
10000.2.a.c.1.2 2 1.1 even 1 trivial
10000.2.a.l.1.1 2 5.4 even 2