# Properties

 Label 225.4.a.m.1.1 Level $225$ Weight $4$ Character 225.1 Self dual yes Analytic conductor $13.275$ Analytic rank $0$ Dimension $2$ CM no Inner twists $2$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 225.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$13.2754297513$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{10})$$ Defining polynomial: $$x^{2} - 10$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: yes Fricke sign: $$1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$-3.16228$$ of defining polynomial Character $$\chi$$ $$=$$ 225.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-3.16228 q^{2} +2.00000 q^{4} +15.0000 q^{7} +18.9737 q^{8} +O(q^{10})$$ $$q-3.16228 q^{2} +2.00000 q^{4} +15.0000 q^{7} +18.9737 q^{8} -63.2456 q^{11} +35.0000 q^{13} -47.4342 q^{14} -76.0000 q^{16} -88.5438 q^{17} +91.0000 q^{19} +200.000 q^{22} +113.842 q^{23} -110.680 q^{26} +30.0000 q^{28} +63.2456 q^{29} -147.000 q^{31} +88.5438 q^{32} +280.000 q^{34} +370.000 q^{37} -287.767 q^{38} +442.719 q^{41} +335.000 q^{43} -126.491 q^{44} -360.000 q^{46} +177.088 q^{47} -118.000 q^{49} +70.0000 q^{52} -88.5438 q^{53} +284.605 q^{56} -200.000 q^{58} +885.438 q^{59} +427.000 q^{61} +464.855 q^{62} +328.000 q^{64} +15.0000 q^{67} -177.088 q^{68} +63.2456 q^{71} -70.0000 q^{73} -1170.04 q^{74} +182.000 q^{76} -948.683 q^{77} -876.000 q^{79} -1400.00 q^{82} +531.263 q^{83} -1059.36 q^{86} -1200.00 q^{88} +525.000 q^{91} +227.684 q^{92} -560.000 q^{94} -1085.00 q^{97} +373.149 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q + 4 q^{4} + 30 q^{7} + O(q^{10})$$ $$2 q + 4 q^{4} + 30 q^{7} + 70 q^{13} - 152 q^{16} + 182 q^{19} + 400 q^{22} + 60 q^{28} - 294 q^{31} + 560 q^{34} + 740 q^{37} + 670 q^{43} - 720 q^{46} - 236 q^{49} + 140 q^{52} - 400 q^{58} + 854 q^{61} + 656 q^{64} + 30 q^{67} - 140 q^{73} + 364 q^{76} - 1752 q^{79} - 2800 q^{82} - 2400 q^{88} + 1050 q^{91} - 1120 q^{94} - 2170 q^{97} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −3.16228 −1.11803 −0.559017 0.829156i $$-0.688821\pi$$
−0.559017 + 0.829156i $$0.688821\pi$$
$$3$$ 0 0
$$4$$ 2.00000 0.250000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 15.0000 0.809924 0.404962 0.914334i $$-0.367285\pi$$
0.404962 + 0.914334i $$0.367285\pi$$
$$8$$ 18.9737 0.838525
$$9$$ 0 0
$$10$$ 0 0
$$11$$ −63.2456 −1.73357 −0.866784 0.498683i $$-0.833817\pi$$
−0.866784 + 0.498683i $$0.833817\pi$$
$$12$$ 0 0
$$13$$ 35.0000 0.746712 0.373356 0.927688i $$-0.378207\pi$$
0.373356 + 0.927688i $$0.378207\pi$$
$$14$$ −47.4342 −0.905522
$$15$$ 0 0
$$16$$ −76.0000 −1.18750
$$17$$ −88.5438 −1.26324 −0.631618 0.775280i $$-0.717609\pi$$
−0.631618 + 0.775280i $$0.717609\pi$$
$$18$$ 0 0
$$19$$ 91.0000 1.09878 0.549390 0.835566i $$-0.314860\pi$$
0.549390 + 0.835566i $$0.314860\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ 200.000 1.93819
$$23$$ 113.842 1.03207 0.516037 0.856566i $$-0.327407\pi$$
0.516037 + 0.856566i $$0.327407\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −110.680 −0.834849
$$27$$ 0 0
$$28$$ 30.0000 0.202481
$$29$$ 63.2456 0.404979 0.202490 0.979284i $$-0.435097\pi$$
0.202490 + 0.979284i $$0.435097\pi$$
$$30$$ 0 0
$$31$$ −147.000 −0.851677 −0.425838 0.904799i $$-0.640021\pi$$
−0.425838 + 0.904799i $$0.640021\pi$$
$$32$$ 88.5438 0.489140
$$33$$ 0 0
$$34$$ 280.000 1.41234
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 370.000 1.64399 0.821995 0.569495i $$-0.192861\pi$$
0.821995 + 0.569495i $$0.192861\pi$$
$$38$$ −287.767 −1.22847
$$39$$ 0 0
$$40$$ 0 0
$$41$$ 442.719 1.68637 0.843184 0.537625i $$-0.180679\pi$$
0.843184 + 0.537625i $$0.180679\pi$$
$$42$$ 0 0
$$43$$ 335.000 1.18807 0.594035 0.804439i $$-0.297534\pi$$
0.594035 + 0.804439i $$0.297534\pi$$
$$44$$ −126.491 −0.433392
$$45$$ 0 0
$$46$$ −360.000 −1.15389
$$47$$ 177.088 0.549593 0.274797 0.961502i $$-0.411390\pi$$
0.274797 + 0.961502i $$0.411390\pi$$
$$48$$ 0 0
$$49$$ −118.000 −0.344023
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 70.0000 0.186678
$$53$$ −88.5438 −0.229480 −0.114740 0.993396i $$-0.536603\pi$$
−0.114740 + 0.993396i $$0.536603\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ 284.605 0.679142
$$57$$ 0 0
$$58$$ −200.000 −0.452781
$$59$$ 885.438 1.95380 0.976900 0.213698i $$-0.0685508\pi$$
0.976900 + 0.213698i $$0.0685508\pi$$
$$60$$ 0 0
$$61$$ 427.000 0.896258 0.448129 0.893969i $$-0.352090\pi$$
0.448129 + 0.893969i $$0.352090\pi$$
$$62$$ 464.855 0.952204
$$63$$ 0 0
$$64$$ 328.000 0.640625
$$65$$ 0 0
$$66$$ 0 0
$$67$$ 15.0000 0.0273514 0.0136757 0.999906i $$-0.495647\pi$$
0.0136757 + 0.999906i $$0.495647\pi$$
$$68$$ −177.088 −0.315809
$$69$$ 0 0
$$70$$ 0 0
$$71$$ 63.2456 0.105716 0.0528582 0.998602i $$-0.483167\pi$$
0.0528582 + 0.998602i $$0.483167\pi$$
$$72$$ 0 0
$$73$$ −70.0000 −0.112231 −0.0561156 0.998424i $$-0.517872\pi$$
−0.0561156 + 0.998424i $$0.517872\pi$$
$$74$$ −1170.04 −1.83804
$$75$$ 0 0
$$76$$ 182.000 0.274695
$$77$$ −948.683 −1.40406
$$78$$ 0 0
$$79$$ −876.000 −1.24757 −0.623783 0.781598i $$-0.714405\pi$$
−0.623783 + 0.781598i $$0.714405\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ −1400.00 −1.88542
$$83$$ 531.263 0.702574 0.351287 0.936268i $$-0.385744\pi$$
0.351287 + 0.936268i $$0.385744\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ −1059.36 −1.32830
$$87$$ 0 0
$$88$$ −1200.00 −1.45364
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 525.000 0.604780
$$92$$ 227.684 0.258018
$$93$$ 0 0
$$94$$ −560.000 −0.614464
$$95$$ 0 0
$$96$$ 0 0
$$97$$ −1085.00 −1.13572 −0.567861 0.823124i $$-0.692229\pi$$
−0.567861 + 0.823124i $$0.692229\pi$$
$$98$$ 373.149 0.384630
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −1328.16 −1.30848 −0.654240 0.756287i $$-0.727011\pi$$
−0.654240 + 0.756287i $$0.727011\pi$$
$$102$$ 0 0
$$103$$ 1540.00 1.47321 0.736605 0.676323i $$-0.236428\pi$$
0.736605 + 0.676323i $$0.236428\pi$$
$$104$$ 664.078 0.626137
$$105$$ 0 0
$$106$$ 280.000 0.256566
$$107$$ −796.894 −0.719987 −0.359994 0.932955i $$-0.617221\pi$$
−0.359994 + 0.932955i $$0.617221\pi$$
$$108$$ 0 0
$$109$$ −811.000 −0.712658 −0.356329 0.934361i $$-0.615972\pi$$
−0.356329 + 0.934361i $$0.615972\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ −1140.00 −0.961785
$$113$$ 1619.09 1.34788 0.673942 0.738785i $$-0.264600\pi$$
0.673942 + 0.738785i $$0.264600\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 126.491 0.101245
$$117$$ 0 0
$$118$$ −2800.00 −2.18441
$$119$$ −1328.16 −1.02313
$$120$$ 0 0
$$121$$ 2669.00 2.00526
$$122$$ −1350.29 −1.00205
$$123$$ 0 0
$$124$$ −294.000 −0.212919
$$125$$ 0 0
$$126$$ 0 0
$$127$$ 260.000 0.181664 0.0908318 0.995866i $$-0.471047\pi$$
0.0908318 + 0.995866i $$0.471047\pi$$
$$128$$ −1745.58 −1.20538
$$129$$ 0 0
$$130$$ 0 0
$$131$$ 1328.16 0.885814 0.442907 0.896568i $$-0.353947\pi$$
0.442907 + 0.896568i $$0.353947\pi$$
$$132$$ 0 0
$$133$$ 1365.00 0.889929
$$134$$ −47.4342 −0.0305798
$$135$$ 0 0
$$136$$ −1680.00 −1.05926
$$137$$ 2656.31 1.65653 0.828263 0.560339i $$-0.189329\pi$$
0.828263 + 0.560339i $$0.189329\pi$$
$$138$$ 0 0
$$139$$ −784.000 −0.478403 −0.239201 0.970970i $$-0.576886\pi$$
−0.239201 + 0.970970i $$0.576886\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ −200.000 −0.118195
$$143$$ −2213.59 −1.29448
$$144$$ 0 0
$$145$$ 0 0
$$146$$ 221.359 0.125478
$$147$$ 0 0
$$148$$ 740.000 0.410997
$$149$$ 2150.35 1.18230 0.591152 0.806560i $$-0.298673\pi$$
0.591152 + 0.806560i $$0.298673\pi$$
$$150$$ 0 0
$$151$$ −797.000 −0.429529 −0.214765 0.976666i $$-0.568898\pi$$
−0.214765 + 0.976666i $$0.568898\pi$$
$$152$$ 1726.60 0.921356
$$153$$ 0 0
$$154$$ 3000.00 1.56978
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 1225.00 0.622711 0.311356 0.950293i $$-0.399217\pi$$
0.311356 + 0.950293i $$0.399217\pi$$
$$158$$ 2770.16 1.39482
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 1707.63 0.835901
$$162$$ 0 0
$$163$$ −275.000 −0.132145 −0.0660726 0.997815i $$-0.521047\pi$$
−0.0660726 + 0.997815i $$0.521047\pi$$
$$164$$ 885.438 0.421592
$$165$$ 0 0
$$166$$ −1680.00 −0.785502
$$167$$ −265.631 −0.123085 −0.0615424 0.998104i $$-0.519602\pi$$
−0.0615424 + 0.998104i $$0.519602\pi$$
$$168$$ 0 0
$$169$$ −972.000 −0.442421
$$170$$ 0 0
$$171$$ 0 0
$$172$$ 670.000 0.297018
$$173$$ −3984.47 −1.75106 −0.875531 0.483162i $$-0.839488\pi$$
−0.875531 + 0.483162i $$0.839488\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 4806.66 2.05861
$$177$$ 0 0
$$178$$ 0 0
$$179$$ −1770.88 −0.739449 −0.369725 0.929141i $$-0.620548\pi$$
−0.369725 + 0.929141i $$0.620548\pi$$
$$180$$ 0 0
$$181$$ −1687.00 −0.692783 −0.346391 0.938090i $$-0.612593\pi$$
−0.346391 + 0.938090i $$0.612593\pi$$
$$182$$ −1660.20 −0.676164
$$183$$ 0 0
$$184$$ 2160.00 0.865420
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 5600.00 2.18991
$$188$$ 354.175 0.137398
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −2213.59 −0.838587 −0.419293 0.907851i $$-0.637722\pi$$
−0.419293 + 0.907851i $$0.637722\pi$$
$$192$$ 0 0
$$193$$ 3625.00 1.35199 0.675993 0.736908i $$-0.263715\pi$$
0.675993 + 0.736908i $$0.263715\pi$$
$$194$$ 3431.07 1.26978
$$195$$ 0 0
$$196$$ −236.000 −0.0860058
$$197$$ 1745.58 0.631306 0.315653 0.948875i $$-0.397776\pi$$
0.315653 + 0.948875i $$0.397776\pi$$
$$198$$ 0 0
$$199$$ 3199.00 1.13955 0.569777 0.821800i $$-0.307030\pi$$
0.569777 + 0.821800i $$0.307030\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 4200.00 1.46293
$$203$$ 948.683 0.328003
$$204$$ 0 0
$$205$$ 0 0
$$206$$ −4869.91 −1.64710
$$207$$ 0 0
$$208$$ −2660.00 −0.886720
$$209$$ −5755.35 −1.90481
$$210$$ 0 0
$$211$$ 887.000 0.289401 0.144700 0.989476i $$-0.453778\pi$$
0.144700 + 0.989476i $$0.453778\pi$$
$$212$$ −177.088 −0.0573699
$$213$$ 0 0
$$214$$ 2520.00 0.804970
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −2205.00 −0.689793
$$218$$ 2564.61 0.796776
$$219$$ 0 0
$$220$$ 0 0
$$221$$ −3099.03 −0.943274
$$222$$ 0 0
$$223$$ 3535.00 1.06153 0.530765 0.847519i $$-0.321905\pi$$
0.530765 + 0.847519i $$0.321905\pi$$
$$224$$ 1328.16 0.396166
$$225$$ 0 0
$$226$$ −5120.00 −1.50698
$$227$$ 88.5438 0.0258892 0.0129446 0.999916i $$-0.495879\pi$$
0.0129446 + 0.999916i $$0.495879\pi$$
$$228$$ 0 0
$$229$$ −5019.00 −1.44832 −0.724159 0.689633i $$-0.757772\pi$$
−0.724159 + 0.689633i $$0.757772\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ 1200.00 0.339586
$$233$$ 3605.00 1.01361 0.506805 0.862061i $$-0.330826\pi$$
0.506805 + 0.862061i $$0.330826\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ 1770.88 0.488450
$$237$$ 0 0
$$238$$ 4200.00 1.14389
$$239$$ −442.719 −0.119821 −0.0599103 0.998204i $$-0.519081\pi$$
−0.0599103 + 0.998204i $$0.519081\pi$$
$$240$$ 0 0
$$241$$ −623.000 −0.166518 −0.0832592 0.996528i $$-0.526533\pi$$
−0.0832592 + 0.996528i $$0.526533\pi$$
$$242$$ −8440.12 −2.24195
$$243$$ 0 0
$$244$$ 854.000 0.224065
$$245$$ 0 0
$$246$$ 0 0
$$247$$ 3185.00 0.820472
$$248$$ −2789.13 −0.714153
$$249$$ 0 0
$$250$$ 0 0
$$251$$ −2656.31 −0.667988 −0.333994 0.942575i $$-0.608397\pi$$
−0.333994 + 0.942575i $$0.608397\pi$$
$$252$$ 0 0
$$253$$ −7200.00 −1.78917
$$254$$ −822.192 −0.203106
$$255$$ 0 0
$$256$$ 2896.00 0.707031
$$257$$ −3718.84 −0.902626 −0.451313 0.892366i $$-0.649044\pi$$
−0.451313 + 0.892366i $$0.649044\pi$$
$$258$$ 0 0
$$259$$ 5550.00 1.33151
$$260$$ 0 0
$$261$$ 0 0
$$262$$ −4200.00 −0.990370
$$263$$ 2896.65 0.679144 0.339572 0.940580i $$-0.389718\pi$$
0.339572 + 0.940580i $$0.389718\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ −4316.51 −0.994970
$$267$$ 0 0
$$268$$ 30.0000 0.00683784
$$269$$ −442.719 −0.100346 −0.0501729 0.998741i $$-0.515977\pi$$
−0.0501729 + 0.998741i $$0.515977\pi$$
$$270$$ 0 0
$$271$$ −308.000 −0.0690394 −0.0345197 0.999404i $$-0.510990\pi$$
−0.0345197 + 0.999404i $$0.510990\pi$$
$$272$$ 6729.33 1.50009
$$273$$ 0 0
$$274$$ −8400.00 −1.85205
$$275$$ 0 0
$$276$$ 0 0
$$277$$ −7485.00 −1.62357 −0.811787 0.583953i $$-0.801505\pi$$
−0.811787 + 0.583953i $$0.801505\pi$$
$$278$$ 2479.23 0.534871
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 948.683 0.201401 0.100701 0.994917i $$-0.467892\pi$$
0.100701 + 0.994917i $$0.467892\pi$$
$$282$$ 0 0
$$283$$ 525.000 0.110276 0.0551378 0.998479i $$-0.482440\pi$$
0.0551378 + 0.998479i $$0.482440\pi$$
$$284$$ 126.491 0.0264291
$$285$$ 0 0
$$286$$ 7000.00 1.44727
$$287$$ 6640.78 1.36583
$$288$$ 0 0
$$289$$ 2927.00 0.595766
$$290$$ 0 0
$$291$$ 0 0
$$292$$ −140.000 −0.0280578
$$293$$ −6375.15 −1.27113 −0.635564 0.772048i $$-0.719232\pi$$
−0.635564 + 0.772048i $$0.719232\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ 7020.26 1.37853
$$297$$ 0 0
$$298$$ −6800.00 −1.32186
$$299$$ 3984.47 0.770662
$$300$$ 0 0
$$301$$ 5025.00 0.962246
$$302$$ 2520.34 0.480228
$$303$$ 0 0
$$304$$ −6916.00 −1.30480
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −2695.00 −0.501016 −0.250508 0.968115i $$-0.580598\pi$$
−0.250508 + 0.968115i $$0.580598\pi$$
$$308$$ −1897.37 −0.351015
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 5312.63 0.968654 0.484327 0.874887i $$-0.339064\pi$$
0.484327 + 0.874887i $$0.339064\pi$$
$$312$$ 0 0
$$313$$ −2555.00 −0.461397 −0.230698 0.973025i $$-0.574101\pi$$
−0.230698 + 0.973025i $$0.574101\pi$$
$$314$$ −3873.79 −0.696212
$$315$$ 0 0
$$316$$ −1752.00 −0.311891
$$317$$ −7462.98 −1.32228 −0.661140 0.750263i $$-0.729927\pi$$
−0.661140 + 0.750263i $$0.729927\pi$$
$$318$$ 0 0
$$319$$ −4000.00 −0.702060
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −5400.00 −0.934566
$$323$$ −8057.48 −1.38802
$$324$$ 0 0
$$325$$ 0 0
$$326$$ 869.626 0.147743
$$327$$ 0 0
$$328$$ 8400.00 1.41406
$$329$$ 2656.31 0.445129
$$330$$ 0 0
$$331$$ −5088.00 −0.844900 −0.422450 0.906386i $$-0.638830\pi$$
−0.422450 + 0.906386i $$0.638830\pi$$
$$332$$ 1062.53 0.175644
$$333$$ 0 0
$$334$$ 840.000 0.137613
$$335$$ 0 0
$$336$$ 0 0
$$337$$ −11485.0 −1.85646 −0.928231 0.372004i $$-0.878671\pi$$
−0.928231 + 0.372004i $$0.878671\pi$$
$$338$$ 3073.73 0.494642
$$339$$ 0 0
$$340$$ 0 0
$$341$$ 9297.10 1.47644
$$342$$ 0 0
$$343$$ −6915.00 −1.08856
$$344$$ 6356.18 0.996227
$$345$$ 0 0
$$346$$ 12600.0 1.95775
$$347$$ 6400.45 0.990185 0.495092 0.868840i $$-0.335134\pi$$
0.495092 + 0.868840i $$0.335134\pi$$
$$348$$ 0 0
$$349$$ −9674.00 −1.48377 −0.741887 0.670525i $$-0.766069\pi$$
−0.741887 + 0.670525i $$0.766069\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ −5600.00 −0.847957
$$353$$ 5578.26 0.841078 0.420539 0.907274i $$-0.361841\pi$$
0.420539 + 0.907274i $$0.361841\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 5600.00 0.826730
$$359$$ 8917.62 1.31101 0.655507 0.755189i $$-0.272455\pi$$
0.655507 + 0.755189i $$0.272455\pi$$
$$360$$ 0 0
$$361$$ 1422.00 0.207319
$$362$$ 5334.76 0.774555
$$363$$ 0 0
$$364$$ 1050.00 0.151195
$$365$$ 0 0
$$366$$ 0 0
$$367$$ 8505.00 1.20969 0.604847 0.796342i $$-0.293234\pi$$
0.604847 + 0.796342i $$0.293234\pi$$
$$368$$ −8651.99 −1.22559
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −1328.16 −0.185861
$$372$$ 0 0
$$373$$ −6775.00 −0.940472 −0.470236 0.882541i $$-0.655831\pi$$
−0.470236 + 0.882541i $$0.655831\pi$$
$$374$$ −17708.8 −2.44839
$$375$$ 0 0
$$376$$ 3360.00 0.460848
$$377$$ 2213.59 0.302403
$$378$$ 0 0
$$379$$ 9241.00 1.25245 0.626225 0.779643i $$-0.284599\pi$$
0.626225 + 0.779643i $$0.284599\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 7000.00 0.937568
$$383$$ 9031.46 1.20493 0.602463 0.798147i $$-0.294186\pi$$
0.602463 + 0.798147i $$0.294186\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ −11463.3 −1.51157
$$387$$ 0 0
$$388$$ −2170.00 −0.283931
$$389$$ 4363.94 0.568794 0.284397 0.958707i $$-0.408207\pi$$
0.284397 + 0.958707i $$0.408207\pi$$
$$390$$ 0 0
$$391$$ −10080.0 −1.30375
$$392$$ −2238.89 −0.288472
$$393$$ 0 0
$$394$$ −5520.00 −0.705821
$$395$$ 0 0
$$396$$ 0 0
$$397$$ −4795.00 −0.606182 −0.303091 0.952962i $$-0.598019\pi$$
−0.303091 + 0.952962i $$0.598019\pi$$
$$398$$ −10116.1 −1.27406
$$399$$ 0 0
$$400$$ 0 0
$$401$$ −7020.26 −0.874252 −0.437126 0.899400i $$-0.644004\pi$$
−0.437126 + 0.899400i $$0.644004\pi$$
$$402$$ 0 0
$$403$$ −5145.00 −0.635957
$$404$$ −2656.31 −0.327120
$$405$$ 0 0
$$406$$ −3000.00 −0.366718
$$407$$ −23400.9 −2.84997
$$408$$ 0 0
$$409$$ 6881.00 0.831891 0.415946 0.909389i $$-0.363451\pi$$
0.415946 + 0.909389i $$0.363451\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ 3080.00 0.368303
$$413$$ 13281.6 1.58243
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 3099.03 0.365247
$$417$$ 0 0
$$418$$ 18200.0 2.12964
$$419$$ −12838.8 −1.49694 −0.748471 0.663167i $$-0.769212\pi$$
−0.748471 + 0.663167i $$0.769212\pi$$
$$420$$ 0 0
$$421$$ 6418.00 0.742979 0.371490 0.928437i $$-0.378847\pi$$
0.371490 + 0.928437i $$0.378847\pi$$
$$422$$ −2804.94 −0.323560
$$423$$ 0 0
$$424$$ −1680.00 −0.192425
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 6405.00 0.725901
$$428$$ −1593.79 −0.179997
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −1328.16 −0.148434 −0.0742170 0.997242i $$-0.523646\pi$$
−0.0742170 + 0.997242i $$0.523646\pi$$
$$432$$ 0 0
$$433$$ −8155.00 −0.905091 −0.452545 0.891741i $$-0.649484\pi$$
−0.452545 + 0.891741i $$0.649484\pi$$
$$434$$ 6972.82 0.771212
$$435$$ 0 0
$$436$$ −1622.00 −0.178164
$$437$$ 10359.6 1.13402
$$438$$ 0 0
$$439$$ 14749.0 1.60349 0.801744 0.597667i $$-0.203906\pi$$
0.801744 + 0.597667i $$0.203906\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ 9800.00 1.05461
$$443$$ 11156.5 1.19653 0.598264 0.801299i $$-0.295857\pi$$
0.598264 + 0.801299i $$0.295857\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −11178.7 −1.18683
$$447$$ 0 0
$$448$$ 4920.00 0.518857
$$449$$ 9233.85 0.970540 0.485270 0.874364i $$-0.338721\pi$$
0.485270 + 0.874364i $$0.338721\pi$$
$$450$$ 0 0
$$451$$ −28000.0 −2.92343
$$452$$ 3238.17 0.336971
$$453$$ 0 0
$$454$$ −280.000 −0.0289450
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −1030.00 −0.105430 −0.0527148 0.998610i $$-0.516787\pi$$
−0.0527148 + 0.998610i $$0.516787\pi$$
$$458$$ 15871.5 1.61927
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −9297.10 −0.939282 −0.469641 0.882858i $$-0.655617\pi$$
−0.469641 + 0.882858i $$0.655617\pi$$
$$462$$ 0 0
$$463$$ 11940.0 1.19849 0.599243 0.800567i $$-0.295468\pi$$
0.599243 + 0.800567i $$0.295468\pi$$
$$464$$ −4806.66 −0.480913
$$465$$ 0 0
$$466$$ −11400.0 −1.13325
$$467$$ −12130.5 −1.20200 −0.600998 0.799250i $$-0.705230\pi$$
−0.600998 + 0.799250i $$0.705230\pi$$
$$468$$ 0 0
$$469$$ 225.000 0.0221525
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 16800.0 1.63831
$$473$$ −21187.3 −2.05960
$$474$$ 0 0
$$475$$ 0 0
$$476$$ −2656.31 −0.255781
$$477$$ 0 0
$$478$$ 1400.00 0.133963
$$479$$ 2656.31 0.253382 0.126691 0.991942i $$-0.459564\pi$$
0.126691 + 0.991942i $$0.459564\pi$$
$$480$$ 0 0
$$481$$ 12950.0 1.22759
$$482$$ 1970.10 0.186173
$$483$$ 0 0
$$484$$ 5338.00 0.501315
$$485$$ 0 0
$$486$$ 0 0
$$487$$ 14125.0 1.31430 0.657151 0.753759i $$-0.271761\pi$$
0.657151 + 0.753759i $$0.271761\pi$$
$$488$$ 8101.76 0.751535
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 4047.72 0.372038 0.186019 0.982546i $$-0.440441\pi$$
0.186019 + 0.982546i $$0.440441\pi$$
$$492$$ 0 0
$$493$$ −5600.00 −0.511585
$$494$$ −10071.9 −0.917316
$$495$$ 0 0
$$496$$ 11172.0 1.01137
$$497$$ 948.683 0.0856223
$$498$$ 0 0
$$499$$ 1279.00 0.114741 0.0573706 0.998353i $$-0.481728\pi$$
0.0573706 + 0.998353i $$0.481728\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ 8400.00 0.746833
$$503$$ −16380.6 −1.45204 −0.726019 0.687675i $$-0.758631\pi$$
−0.726019 + 0.687675i $$0.758631\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ 22768.4 2.00035
$$507$$ 0 0
$$508$$ 520.000 0.0454159
$$509$$ −4427.19 −0.385524 −0.192762 0.981246i $$-0.561745\pi$$
−0.192762 + 0.981246i $$0.561745\pi$$
$$510$$ 0 0
$$511$$ −1050.00 −0.0908988
$$512$$ 4806.66 0.414895
$$513$$ 0 0
$$514$$ 11760.0 1.00917
$$515$$ 0 0
$$516$$ 0 0
$$517$$ −11200.0 −0.952757
$$518$$ −17550.6 −1.48867
$$519$$ 0 0
$$520$$ 0 0
$$521$$ 11068.0 0.930704 0.465352 0.885126i $$-0.345928\pi$$
0.465352 + 0.885126i $$0.345928\pi$$
$$522$$ 0 0
$$523$$ 10745.0 0.898367 0.449184 0.893439i $$-0.351715\pi$$
0.449184 + 0.893439i $$0.351715\pi$$
$$524$$ 2656.31 0.221453
$$525$$ 0 0
$$526$$ −9160.00 −0.759306
$$527$$ 13015.9 1.07587
$$528$$ 0 0
$$529$$ 793.000 0.0651763
$$530$$ 0 0
$$531$$ 0 0
$$532$$ 2730.00 0.222482
$$533$$ 15495.2 1.25923
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 284.605 0.0229348
$$537$$ 0 0
$$538$$ 1400.00 0.112190
$$539$$ 7462.98 0.596388
$$540$$ 0 0
$$541$$ −9423.00 −0.748847 −0.374424 0.927258i $$-0.622159\pi$$
−0.374424 + 0.927258i $$0.622159\pi$$
$$542$$ 973.982 0.0771884
$$543$$ 0 0
$$544$$ −7840.00 −0.617899
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 9080.00 0.709749 0.354875 0.934914i $$-0.384524\pi$$
0.354875 + 0.934914i $$0.384524\pi$$
$$548$$ 5312.63 0.414132
$$549$$ 0 0
$$550$$ 0 0
$$551$$ 5755.35 0.444984
$$552$$ 0 0
$$553$$ −13140.0 −1.01043
$$554$$ 23669.6 1.81521
$$555$$ 0 0
$$556$$ −1568.00 −0.119601
$$557$$ 8348.41 0.635069 0.317535 0.948247i $$-0.397145\pi$$
0.317535 + 0.948247i $$0.397145\pi$$
$$558$$ 0 0
$$559$$ 11725.0 0.887146
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −3000.00 −0.225173
$$563$$ −7703.31 −0.576653 −0.288327 0.957532i $$-0.593099\pi$$
−0.288327 + 0.957532i $$0.593099\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −1660.20 −0.123292
$$567$$ 0 0
$$568$$ 1200.00 0.0886459
$$569$$ 6261.31 0.461314 0.230657 0.973035i $$-0.425912\pi$$
0.230657 + 0.973035i $$0.425912\pi$$
$$570$$ 0 0
$$571$$ 587.000 0.0430213 0.0215107 0.999769i $$-0.493152\pi$$
0.0215107 + 0.999769i $$0.493152\pi$$
$$572$$ −4427.19 −0.323619
$$573$$ 0 0
$$574$$ −21000.0 −1.52704
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −15995.0 −1.15404 −0.577020 0.816730i $$-0.695784\pi$$
−0.577020 + 0.816730i $$0.695784\pi$$
$$578$$ −9255.99 −0.666087
$$579$$ 0 0
$$580$$ 0 0
$$581$$ 7968.94 0.569032
$$582$$ 0 0
$$583$$ 5600.00 0.397819
$$584$$ −1328.16 −0.0941088
$$585$$ 0 0
$$586$$ 20160.0 1.42116
$$587$$ 20453.6 1.43818 0.719089 0.694918i $$-0.244559\pi$$
0.719089 + 0.694918i $$0.244559\pi$$
$$588$$ 0 0
$$589$$ −13377.0 −0.935806
$$590$$ 0 0
$$591$$ 0 0
$$592$$ −28120.0 −1.95224
$$593$$ −11599.2 −0.803244 −0.401622 0.915806i $$-0.631553\pi$$
−0.401622 + 0.915806i $$0.631553\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ 4300.70 0.295576
$$597$$ 0 0
$$598$$ −12600.0 −0.861626
$$599$$ −19100.2 −1.30286 −0.651428 0.758710i $$-0.725830\pi$$
−0.651428 + 0.758710i $$0.725830\pi$$
$$600$$ 0 0
$$601$$ 3143.00 0.213321 0.106660 0.994296i $$-0.465984\pi$$
0.106660 + 0.994296i $$0.465984\pi$$
$$602$$ −15890.4 −1.07582
$$603$$ 0 0
$$604$$ −1594.00 −0.107382
$$605$$ 0 0
$$606$$ 0 0
$$607$$ 2660.00 0.177868 0.0889342 0.996038i $$-0.471654\pi$$
0.0889342 + 0.996038i $$0.471654\pi$$
$$608$$ 8057.48 0.537457
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 6198.06 0.410388
$$612$$ 0 0
$$613$$ −6670.00 −0.439476 −0.219738 0.975559i $$-0.570520\pi$$
−0.219738 + 0.975559i $$0.570520\pi$$
$$614$$ 8522.34 0.560152
$$615$$ 0 0
$$616$$ −18000.0 −1.17734
$$617$$ −11042.7 −0.720521 −0.360260 0.932852i $$-0.617312\pi$$
−0.360260 + 0.932852i $$0.617312\pi$$
$$618$$ 0 0
$$619$$ 5579.00 0.362260 0.181130 0.983459i $$-0.442025\pi$$
0.181130 + 0.983459i $$0.442025\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −16800.0 −1.08299
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ 8079.62 0.515857
$$627$$ 0 0
$$628$$ 2450.00 0.155678
$$629$$ −32761.2 −2.07675
$$630$$ 0 0
$$631$$ 23617.0 1.48998 0.744990 0.667075i $$-0.232454\pi$$
0.744990 + 0.667075i $$0.232454\pi$$
$$632$$ −16620.9 −1.04612
$$633$$ 0 0
$$634$$ 23600.0 1.47835
$$635$$ 0 0
$$636$$ 0 0
$$637$$ −4130.00 −0.256886
$$638$$ 12649.1 0.784926
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −31496.3 −1.94076 −0.970381 0.241579i $$-0.922335\pi$$
−0.970381 + 0.241579i $$0.922335\pi$$
$$642$$ 0 0
$$643$$ −14280.0 −0.875814 −0.437907 0.899020i $$-0.644280\pi$$
−0.437907 + 0.899020i $$0.644280\pi$$
$$644$$ 3415.26 0.208975
$$645$$ 0 0
$$646$$ 25480.0 1.55185
$$647$$ 12396.1 0.753234 0.376617 0.926369i $$-0.377087\pi$$
0.376617 + 0.926369i $$0.377087\pi$$
$$648$$ 0 0
$$649$$ −56000.0 −3.38705
$$650$$ 0 0
$$651$$ 0 0
$$652$$ −550.000 −0.0330363
$$653$$ 12042.0 0.721651 0.360825 0.932633i $$-0.382495\pi$$
0.360825 + 0.932633i $$0.382495\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −33646.6 −2.00256
$$657$$ 0 0
$$658$$ −8400.00 −0.497669
$$659$$ −5755.35 −0.340207 −0.170104 0.985426i $$-0.554410\pi$$
−0.170104 + 0.985426i $$0.554410\pi$$
$$660$$ 0 0
$$661$$ 25438.0 1.49686 0.748429 0.663215i $$-0.230808\pi$$
0.748429 + 0.663215i $$0.230808\pi$$
$$662$$ 16089.7 0.944626
$$663$$ 0 0
$$664$$ 10080.0 0.589126
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 7200.00 0.417969
$$668$$ −531.263 −0.0307712
$$669$$ 0 0
$$670$$ 0 0
$$671$$ −27005.9 −1.55372
$$672$$ 0 0
$$673$$ 15150.0 0.867741 0.433870 0.900975i $$-0.357148\pi$$
0.433870 + 0.900975i $$0.357148\pi$$
$$674$$ 36318.8 2.07559
$$675$$ 0 0
$$676$$ −1944.00 −0.110605
$$677$$ −20984.9 −1.19131 −0.595653 0.803242i $$-0.703107\pi$$
−0.595653 + 0.803242i $$0.703107\pi$$
$$678$$ 0 0
$$679$$ −16275.0 −0.919849
$$680$$ 0 0
$$681$$ 0 0
$$682$$ −29400.0 −1.65071
$$683$$ 1973.26 0.110549 0.0552743 0.998471i $$-0.482397\pi$$
0.0552743 + 0.998471i $$0.482397\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ 21867.2 1.21704
$$687$$ 0 0
$$688$$ −25460.0 −1.41083
$$689$$ −3099.03 −0.171355
$$690$$ 0 0
$$691$$ −20552.0 −1.13145 −0.565727 0.824592i $$-0.691404\pi$$
−0.565727 + 0.824592i $$0.691404\pi$$
$$692$$ −7968.94 −0.437765
$$693$$ 0 0
$$694$$ −20240.0 −1.10706
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −39200.0 −2.13028
$$698$$ 30591.9 1.65891
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 8411.66 0.453215 0.226608 0.973986i $$-0.427236\pi$$
0.226608 + 0.973986i $$0.427236\pi$$
$$702$$ 0 0
$$703$$ 33670.0 1.80638
$$704$$ −20744.5 −1.11057
$$705$$ 0 0
$$706$$ −17640.0 −0.940354
$$707$$ −19922.3 −1.05977
$$708$$ 0 0
$$709$$ 17281.0 0.915376 0.457688 0.889113i $$-0.348678\pi$$
0.457688 + 0.889113i $$0.348678\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ −16734.8 −0.878993
$$714$$ 0 0
$$715$$ 0 0
$$716$$ −3541.75 −0.184862
$$717$$ 0 0
$$718$$ −28200.0 −1.46576
$$719$$ 9297.10 0.482230 0.241115 0.970497i $$-0.422487\pi$$
0.241115 + 0.970497i $$0.422487\pi$$
$$720$$ 0 0
$$721$$ 23100.0 1.19319
$$722$$ −4496.76 −0.231790
$$723$$ 0 0
$$724$$ −3374.00 −0.173196
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 30415.0 1.55162 0.775811 0.630965i $$-0.217341\pi$$
0.775811 + 0.630965i $$0.217341\pi$$
$$728$$ 9961.17 0.507123
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −29662.2 −1.50081
$$732$$ 0 0
$$733$$ 18550.0 0.934734 0.467367 0.884063i $$-0.345203\pi$$
0.467367 + 0.884063i $$0.345203\pi$$
$$734$$ −26895.2 −1.35248
$$735$$ 0 0
$$736$$ 10080.0 0.504828
$$737$$ −948.683 −0.0474155
$$738$$ 0 0
$$739$$ 24016.0 1.19546 0.597729 0.801699i $$-0.296070\pi$$
0.597729 + 0.801699i $$0.296070\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 4200.00 0.207799
$$743$$ 12396.1 0.612072 0.306036 0.952020i $$-0.400997\pi$$
0.306036 + 0.952020i $$0.400997\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 21424.4 1.05148
$$747$$ 0 0
$$748$$ 11200.0 0.547477
$$749$$ −11953.4 −0.583135
$$750$$ 0 0
$$751$$ −7772.00 −0.377636 −0.188818 0.982012i $$-0.560466\pi$$
−0.188818 + 0.982012i $$0.560466\pi$$
$$752$$ −13458.7 −0.652642
$$753$$ 0 0
$$754$$ −7000.00 −0.338097
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −3865.00 −0.185569 −0.0927846 0.995686i $$-0.529577\pi$$
−0.0927846 + 0.995686i $$0.529577\pi$$
$$758$$ −29222.6 −1.40028
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 15937.9 0.759195 0.379598 0.925152i $$-0.376062\pi$$
0.379598 + 0.925152i $$0.376062\pi$$
$$762$$ 0 0
$$763$$ −12165.0 −0.577199
$$764$$ −4427.19 −0.209647
$$765$$ 0 0
$$766$$ −28560.0 −1.34715
$$767$$ 30990.3 1.45893
$$768$$ 0 0
$$769$$ −7441.00 −0.348933 −0.174466 0.984663i $$-0.555820\pi$$
−0.174466 + 0.984663i $$0.555820\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ 7250.00 0.337996
$$773$$ −25766.2 −1.19890 −0.599448 0.800413i $$-0.704613\pi$$
−0.599448 + 0.800413i $$0.704613\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −20586.4 −0.952332
$$777$$ 0 0
$$778$$ −13800.0 −0.635931
$$779$$ 40287.4 1.85295
$$780$$ 0 0
$$781$$ −4000.00 −0.183267
$$782$$ 31875.8 1.45764
$$783$$ 0 0
$$784$$ 8968.00 0.408528
$$785$$ 0 0
$$786$$ 0 0
$$787$$ 6125.00 0.277424 0.138712 0.990333i $$-0.455704\pi$$
0.138712 + 0.990333i $$0.455704\pi$$
$$788$$ 3491.15 0.157826
$$789$$ 0 0
$$790$$ 0 0
$$791$$ 24286.3 1.09168
$$792$$ 0 0
$$793$$ 14945.0 0.669247
$$794$$ 15163.1 0.677732
$$795$$ 0 0
$$796$$ 6398.00 0.284888
$$797$$ 20010.9 0.889363 0.444681 0.895689i $$-0.353317\pi$$
0.444681 + 0.895689i $$0.353317\pi$$
$$798$$ 0 0
$$799$$ −15680.0 −0.694266
$$800$$ 0 0
$$801$$ 0 0
$$802$$ 22200.0 0.977443
$$803$$ 4427.19 0.194561
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 16269.9 0.711022
$$807$$ 0 0
$$808$$ −25200.0 −1.09719
$$809$$ 2719.56 0.118189 0.0590943 0.998252i $$-0.481179\pi$$
0.0590943 + 0.998252i $$0.481179\pi$$
$$810$$ 0 0
$$811$$ 28623.0 1.23932 0.619661 0.784870i $$-0.287270\pi$$
0.619661 + 0.784870i $$0.287270\pi$$
$$812$$ 1897.37 0.0820006
$$813$$ 0 0
$$814$$ 74000.0 3.18636
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 30485.0 1.30543
$$818$$ −21759.6 −0.930083
$$819$$ 0 0
$$820$$ 0 0
$$821$$ 26563.1 1.12918 0.564592 0.825370i $$-0.309034\pi$$
0.564592 + 0.825370i $$0.309034\pi$$
$$822$$ 0 0
$$823$$ 1135.00 0.0480724 0.0240362 0.999711i $$-0.492348\pi$$
0.0240362 + 0.999711i $$0.492348\pi$$
$$824$$ 29219.4 1.23532
$$825$$ 0 0
$$826$$ −42000.0 −1.76921
$$827$$ −999.280 −0.0420174 −0.0210087 0.999779i $$-0.506688\pi$$
−0.0210087 + 0.999779i $$0.506688\pi$$
$$828$$ 0 0
$$829$$ −1974.00 −0.0827019 −0.0413509 0.999145i $$-0.513166\pi$$
−0.0413509 + 0.999145i $$0.513166\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ 11480.0 0.478362
$$833$$ 10448.2 0.434583
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −11510.7 −0.476203
$$837$$ 0 0
$$838$$ 40600.0 1.67363
$$839$$ −42058.3 −1.73065 −0.865324 0.501213i $$-0.832887\pi$$
−0.865324 + 0.501213i $$0.832887\pi$$
$$840$$ 0 0
$$841$$ −20389.0 −0.835992
$$842$$ −20295.5 −0.830676
$$843$$ 0 0
$$844$$ 1774.00 0.0723502
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 40035.0 1.62411
$$848$$ 6729.33 0.272507
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 42121.5 1.69672
$$852$$ 0 0
$$853$$ 36645.0 1.47093 0.735464 0.677564i $$-0.236964\pi$$
0.735464 + 0.677564i $$0.236964\pi$$
$$854$$ −20254.4 −0.811582
$$855$$ 0 0
$$856$$ −15120.0 −0.603728
$$857$$ 10802.3 0.430573 0.215286 0.976551i $$-0.430931\pi$$
0.215286 + 0.976551i $$0.430931\pi$$
$$858$$ 0 0
$$859$$ 20104.0 0.798533 0.399266 0.916835i $$-0.369265\pi$$
0.399266 + 0.916835i $$0.369265\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 4200.00 0.165954
$$863$$ −2150.35 −0.0848189 −0.0424095 0.999100i $$-0.513503\pi$$
−0.0424095 + 0.999100i $$0.513503\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ 25788.4 1.01192
$$867$$ 0 0
$$868$$ −4410.00 −0.172448
$$869$$ 55403.1 2.16274
$$870$$ 0 0
$$871$$ 525.000 0.0204236
$$872$$ −15387.6 −0.597582
$$873$$ 0 0
$$874$$ −32760.0 −1.26788
$$875$$ 0 0
$$876$$ 0 0
$$877$$ −21975.0 −0.846115 −0.423058 0.906103i $$-0.639043\pi$$
−0.423058 + 0.906103i $$0.639043\pi$$
$$878$$ −46640.4 −1.79275
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 885.438 0.0338606 0.0169303 0.999857i $$-0.494611\pi$$
0.0169303 + 0.999857i $$0.494611\pi$$
$$882$$ 0 0
$$883$$ 34915.0 1.33067 0.665336 0.746544i $$-0.268288\pi$$
0.665336 + 0.746544i $$0.268288\pi$$
$$884$$ −6198.06 −0.235818
$$885$$ 0 0
$$886$$ −35280.0 −1.33776
$$887$$ −49407.4 −1.87028 −0.935140 0.354277i $$-0.884727\pi$$
−0.935140 + 0.354277i $$0.884727\pi$$
$$888$$ 0 0
$$889$$ 3900.00 0.147134
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 7070.00 0.265382
$$893$$ 16115.0 0.603882
$$894$$ 0 0
$$895$$ 0 0
$$896$$ −26183.7 −0.976266
$$897$$ 0 0
$$898$$ −29200.0 −1.08510
$$899$$ −9297.10 −0.344912
$$900$$ 0 0
$$901$$ 7840.00 0.289887
$$902$$ 88543.8 3.26850
$$903$$ 0 0
$$904$$ 30720.0 1.13023
$$905$$ 0 0
$$906$$ 0 0
$$907$$ 17880.0 0.654571 0.327285 0.944926i $$-0.393866\pi$$
0.327285 + 0.944926i $$0.393866\pi$$
$$908$$ 177.088 0.00647231
$$909$$ 0 0
$$910$$ 0 0
$$911$$ −17329.3 −0.630236 −0.315118 0.949053i $$-0.602044\pi$$
−0.315118 + 0.949053i $$0.602044\pi$$
$$912$$ 0 0
$$913$$ −33600.0 −1.21796
$$914$$ 3257.15 0.117874
$$915$$ 0 0
$$916$$ −10038.0 −0.362080
$$917$$ 19922.3 0.717442
$$918$$ 0 0
$$919$$ −25829.0 −0.927117 −0.463558 0.886066i $$-0.653428\pi$$
−0.463558 + 0.886066i $$0.653428\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 29400.0 1.05015
$$923$$ 2213.59 0.0789397
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −37757.6 −1.33995
$$927$$ 0 0
$$928$$ 5600.00 0.198092
$$929$$ 3099.03 0.109447 0.0547233 0.998502i $$-0.482572\pi$$
0.0547233 + 0.998502i $$0.482572\pi$$
$$930$$ 0 0
$$931$$ −10738.0 −0.378006
$$932$$ 7209.99 0.253403
$$933$$ 0 0
$$934$$ 38360.0 1.34387
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 40005.0 1.39478 0.697389 0.716693i $$-0.254345\pi$$
0.697389 + 0.716693i $$0.254345\pi$$
$$938$$ −711.512 −0.0247673
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −30104.9 −1.04292 −0.521462 0.853275i $$-0.674613\pi$$
−0.521462 + 0.853275i $$0.674613\pi$$
$$942$$ 0 0
$$943$$ 50400.0 1.74046
$$944$$ −67293.3 −2.32014
$$945$$ 0 0
$$946$$ 67000.0 2.30270
$$947$$ −43297.9 −1.48574 −0.742868 0.669437i $$-0.766535\pi$$
−0.742868 + 0.669437i $$0.766535\pi$$
$$948$$ 0 0
$$949$$ −2450.00 −0.0838044
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −25200.0 −0.857917
$$953$$ 53682.8 1.82472 0.912360 0.409390i $$-0.134258\pi$$
0.912360 + 0.409390i $$0.134258\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ −885.438 −0.0299551
$$957$$ 0 0
$$958$$ −8400.00 −0.283290
$$959$$ 39844.7 1.34166
$$960$$ 0 0
$$961$$ −8182.00 −0.274647
$$962$$ −40951.5 −1.37248
$$963$$ 0 0
$$964$$ −1246.00 −0.0416296
$$965$$ 0 0
$$966$$ 0 0
$$967$$ −20540.0 −0.683063 −0.341531 0.939870i $$-0.610946\pi$$
−0.341531 + 0.939870i $$0.610946\pi$$
$$968$$ 50640.7 1.68146
$$969$$ 0 0
$$970$$ 0 0
$$971$$ −10625.3 −0.351164 −0.175582 0.984465i $$-0.556181\pi$$
−0.175582 + 0.984465i $$0.556181\pi$$
$$972$$ 0 0
$$973$$ −11760.0 −0.387470
$$974$$ −44667.2 −1.46943
$$975$$ 0 0
$$976$$ −32452.0 −1.06431
$$977$$ −28359.3 −0.928654 −0.464327 0.885664i $$-0.653704\pi$$
−0.464327 + 0.885664i $$0.653704\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −12800.0 −0.415952
$$983$$ −35063.3 −1.13769 −0.568844 0.822446i $$-0.692609\pi$$
−0.568844 + 0.822446i $$0.692609\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ 17708.8 0.571969
$$987$$ 0 0
$$988$$ 6370.00 0.205118
$$989$$ 38137.1 1.22618
$$990$$ 0 0
$$991$$ −18283.0 −0.586053 −0.293027 0.956104i $$-0.594662\pi$$
−0.293027 + 0.956104i $$0.594662\pi$$
$$992$$ −13015.9 −0.416589
$$993$$ 0 0
$$994$$ −3000.00 −0.0957286
$$995$$ 0 0
$$996$$ 0 0
$$997$$ −13230.0 −0.420259 −0.210130 0.977674i $$-0.567389\pi$$
−0.210130 + 0.977674i $$0.567389\pi$$
$$998$$ −4044.55 −0.128285
$$999$$ 0 0
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 225.4.a.m.1.1 yes 2
3.2 odd 2 inner 225.4.a.m.1.2 yes 2
5.2 odd 4 225.4.b.i.199.2 4
5.3 odd 4 225.4.b.i.199.3 4
5.4 even 2 225.4.a.l.1.2 yes 2
15.2 even 4 225.4.b.i.199.4 4
15.8 even 4 225.4.b.i.199.1 4
15.14 odd 2 225.4.a.l.1.1 2

By twisted newform
Twist Min Dim Char Parity Ord Type
225.4.a.l.1.1 2 15.14 odd 2
225.4.a.l.1.2 yes 2 5.4 even 2
225.4.a.m.1.1 yes 2 1.1 even 1 trivial
225.4.a.m.1.2 yes 2 3.2 odd 2 inner
225.4.b.i.199.1 4 15.8 even 4
225.4.b.i.199.2 4 5.2 odd 4
225.4.b.i.199.3 4 5.3 odd 4
225.4.b.i.199.4 4 15.2 even 4