# Properties

 Label 225.4.b.i.199.1 Level $225$ Weight $4$ Character 225.199 Analytic conductor $13.275$ Analytic rank $0$ Dimension $4$ CM no Inner twists $4$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$225 = 3^{2} \cdot 5^{2}$$ Weight: $$k$$ $$=$$ $$4$$ Character orbit: $$[\chi]$$ $$=$$ 225.b (of order $$2$$, degree $$1$$, not minimal)

## Newform invariants

 Self dual: no Analytic conductor: $$13.2754297513$$ Analytic rank: $$0$$ Dimension: $$4$$ Coefficient field: $$\Q(i, \sqrt{10})$$ Defining polynomial: $$x^{4} + 25$$ x^4 + 25 Coefficient ring: $$\Z[a_1, \ldots, a_{11}]$$ Coefficient ring index: $$2\cdot 5^{2}$$ Twist minimal: yes Sato-Tate group: $\mathrm{SU}(2)[C_{2}]$

## Embedding invariants

 Embedding label 199.1 Root $$-1.58114 + 1.58114i$$ of defining polynomial Character $$\chi$$ $$=$$ 225.199 Dual form 225.4.b.i.199.4

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-3.16228i q^{2} -2.00000 q^{4} -15.0000i q^{7} -18.9737i q^{8} +O(q^{10})$$ $$q-3.16228i q^{2} -2.00000 q^{4} -15.0000i q^{7} -18.9737i q^{8} +63.2456 q^{11} +35.0000i q^{13} -47.4342 q^{14} -76.0000 q^{16} -88.5438i q^{17} -91.0000 q^{19} -200.000i q^{22} -113.842i q^{23} +110.680 q^{26} +30.0000i q^{28} +63.2456 q^{29} -147.000 q^{31} +88.5438i q^{32} -280.000 q^{34} -370.000i q^{37} +287.767i q^{38} -442.719 q^{41} +335.000i q^{43} -126.491 q^{44} -360.000 q^{46} +177.088i q^{47} +118.000 q^{49} -70.0000i q^{52} +88.5438i q^{53} -284.605 q^{56} -200.000i q^{58} +885.438 q^{59} +427.000 q^{61} +464.855i q^{62} -328.000 q^{64} -15.0000i q^{67} +177.088i q^{68} -63.2456 q^{71} -70.0000i q^{73} -1170.04 q^{74} +182.000 q^{76} -948.683i q^{77} +876.000 q^{79} +1400.00i q^{82} -531.263i q^{83} +1059.36 q^{86} -1200.00i q^{88} +525.000 q^{91} +227.684i q^{92} +560.000 q^{94} +1085.00i q^{97} -373.149i q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$4 q - 8 q^{4}+O(q^{10})$$ 4 * q - 8 * q^4 $$4 q - 8 q^{4} - 304 q^{16} - 364 q^{19} - 588 q^{31} - 1120 q^{34} - 1440 q^{46} + 472 q^{49} + 1708 q^{61} - 1312 q^{64} + 728 q^{76} + 3504 q^{79} + 2100 q^{91} + 2240 q^{94}+O(q^{100})$$ 4 * q - 8 * q^4 - 304 * q^16 - 364 * q^19 - 588 * q^31 - 1120 * q^34 - 1440 * q^46 + 472 * q^49 + 1708 * q^61 - 1312 * q^64 + 728 * q^76 + 3504 * q^79 + 2100 * q^91 + 2240 * q^94

## Character values

We give the values of $$\chi$$ on generators for $$\left(\mathbb{Z}/225\mathbb{Z}\right)^\times$$.

 $$n$$ $$101$$ $$127$$ $$\chi(n)$$ $$1$$ $$-1$$

## Coefficient data

For each $$n$$ we display the coefficients of the $$q$$-expansion $$a_n$$, the Satake parameters $$\alpha_p$$, and the Satake angles $$\theta_p = \textrm{Arg}(\alpha_p)$$.

Display $$a_p$$ with $$p$$ up to: 50 250 1000 Display $$a_n$$ with $$n$$ up to: 50 250 1000
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ − 3.16228i − 1.11803i −0.829156 0.559017i $$-0.811179\pi$$
0.829156 0.559017i $$-0.188821\pi$$
$$3$$ 0 0
$$4$$ −2.00000 −0.250000
$$5$$ 0 0
$$6$$ 0 0
$$7$$ − 15.0000i − 0.809924i −0.914334 0.404962i $$-0.867285\pi$$
0.914334 0.404962i $$-0.132715\pi$$
$$8$$ − 18.9737i − 0.838525i
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 63.2456 1.73357 0.866784 0.498683i $$-0.166183\pi$$
0.866784 + 0.498683i $$0.166183\pi$$
$$12$$ 0 0
$$13$$ 35.0000i 0.746712i 0.927688 + 0.373356i $$0.121793\pi$$
−0.927688 + 0.373356i $$0.878207\pi$$
$$14$$ −47.4342 −0.905522
$$15$$ 0 0
$$16$$ −76.0000 −1.18750
$$17$$ − 88.5438i − 1.26324i −0.775280 0.631618i $$-0.782391\pi$$
0.775280 0.631618i $$-0.217609\pi$$
$$18$$ 0 0
$$19$$ −91.0000 −1.09878 −0.549390 0.835566i $$-0.685140\pi$$
−0.549390 + 0.835566i $$0.685140\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ − 200.000i − 1.93819i
$$23$$ − 113.842i − 1.03207i −0.856566 0.516037i $$-0.827407\pi$$
0.856566 0.516037i $$-0.172593\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ 110.680 0.834849
$$27$$ 0 0
$$28$$ 30.0000i 0.202481i
$$29$$ 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 $$0.640021\pi$$
$$32$$ 88.5438i 0.489140i
$$33$$ 0 0
$$34$$ −280.000 −1.41234
$$35$$ 0 0
$$36$$ 0 0
$$37$$ − 370.000i − 1.64399i −0.569495 0.821995i $$-0.692861\pi$$
0.569495 0.821995i $$-0.307139\pi$$
$$38$$ 287.767i 1.22847i
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −442.719 −1.68637 −0.843184 0.537625i $$-0.819321\pi$$
−0.843184 + 0.537625i $$0.819321\pi$$
$$42$$ 0 0
$$43$$ 335.000i 1.18807i 0.804439 + 0.594035i $$0.202466\pi$$
−0.804439 + 0.594035i $$0.797534\pi$$
$$44$$ −126.491 −0.433392
$$45$$ 0 0
$$46$$ −360.000 −1.15389
$$47$$ 177.088i 0.549593i 0.961502 + 0.274797i $$0.0886105\pi$$
−0.961502 + 0.274797i $$0.911390\pi$$
$$48$$ 0 0
$$49$$ 118.000 0.344023
$$50$$ 0 0
$$51$$ 0 0
$$52$$ − 70.0000i − 0.186678i
$$53$$ 88.5438i 0.229480i 0.993396 + 0.114740i $$0.0366034\pi$$
−0.993396 + 0.114740i $$0.963397\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −284.605 −0.679142
$$57$$ 0 0
$$58$$ − 200.000i − 0.452781i
$$59$$ 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 $$0.352090\pi$$
$$62$$ 464.855i 0.952204i
$$63$$ 0 0
$$64$$ −328.000 −0.640625
$$65$$ 0 0
$$66$$ 0 0
$$67$$ − 15.0000i − 0.0273514i −0.999906 0.0136757i $$-0.995647\pi$$
0.999906 0.0136757i $$-0.00435324\pi$$
$$68$$ 177.088i 0.315809i
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −63.2456 −0.105716 −0.0528582 0.998602i $$-0.516833\pi$$
−0.0528582 + 0.998602i $$0.516833\pi$$
$$72$$ 0 0
$$73$$ − 70.0000i − 0.112231i −0.998424 0.0561156i $$-0.982128\pi$$
0.998424 0.0561156i $$-0.0178715\pi$$
$$74$$ −1170.04 −1.83804
$$75$$ 0 0
$$76$$ 182.000 0.274695
$$77$$ − 948.683i − 1.40406i
$$78$$ 0 0
$$79$$ 876.000 1.24757 0.623783 0.781598i $$-0.285595\pi$$
0.623783 + 0.781598i $$0.285595\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 1400.00i 1.88542i
$$83$$ − 531.263i − 0.702574i −0.936268 0.351287i $$-0.885744\pi$$
0.936268 0.351287i $$-0.114256\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 1059.36 1.32830
$$87$$ 0 0
$$88$$ − 1200.00i − 1.45364i
$$89$$ 0 0 1.00000i $$-0.5\pi$$
1.00000i $$0.5\pi$$
$$90$$ 0 0
$$91$$ 525.000 0.604780
$$92$$ 227.684i 0.258018i
$$93$$ 0 0
$$94$$ 560.000 0.614464
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 1085.00i 1.13572i 0.823124 + 0.567861i $$0.192229\pi$$
−0.823124 + 0.567861i $$0.807771\pi$$
$$98$$ − 373.149i − 0.384630i
$$99$$ 0 0
$$100$$ 0 0
$$101$$ 1328.16 1.30848 0.654240 0.756287i $$-0.272989\pi$$
0.654240 + 0.756287i $$0.272989\pi$$
$$102$$ 0 0
$$103$$ 1540.00i 1.47321i 0.676323 + 0.736605i $$0.263572\pi$$
−0.676323 + 0.736605i $$0.736428\pi$$
$$104$$ 664.078 0.626137
$$105$$ 0 0
$$106$$ 280.000 0.256566
$$107$$ − 796.894i − 0.719987i −0.932955 0.359994i $$-0.882779\pi$$
0.932955 0.359994i $$-0.117221\pi$$
$$108$$ 0 0
$$109$$ 811.000 0.712658 0.356329 0.934361i $$-0.384028\pi$$
0.356329 + 0.934361i $$0.384028\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 1140.00i 0.961785i
$$113$$ − 1619.09i − 1.34788i −0.738785 0.673942i $$-0.764600\pi$$
0.738785 0.673942i $$-0.235400\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ −126.491 −0.101245
$$117$$ 0 0
$$118$$ − 2800.00i − 2.18441i
$$119$$ −1328.16 −1.02313
$$120$$ 0 0
$$121$$ 2669.00 2.00526
$$122$$ − 1350.29i − 1.00205i
$$123$$ 0 0
$$124$$ 294.000 0.212919
$$125$$ 0 0
$$126$$ 0 0
$$127$$ − 260.000i − 0.181664i −0.995866 0.0908318i $$-0.971047\pi$$
0.995866 0.0908318i $$-0.0289526\pi$$
$$128$$ 1745.58i 1.20538i
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −1328.16 −0.885814 −0.442907 0.896568i $$-0.646053\pi$$
−0.442907 + 0.896568i $$0.646053\pi$$
$$132$$ 0 0
$$133$$ 1365.00i 0.889929i
$$134$$ −47.4342 −0.0305798
$$135$$ 0 0
$$136$$ −1680.00 −1.05926
$$137$$ 2656.31i 1.65653i 0.560339 + 0.828263i $$0.310671\pi$$
−0.560339 + 0.828263i $$0.689329\pi$$
$$138$$ 0 0
$$139$$ 784.000 0.478403 0.239201 0.970970i $$-0.423114\pi$$
0.239201 + 0.970970i $$0.423114\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 200.000i 0.118195i
$$143$$ 2213.59i 1.29448i
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −221.359 −0.125478
$$147$$ 0 0
$$148$$ 740.000i 0.410997i
$$149$$ 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 $$0.568898\pi$$
$$152$$ 1726.60i 0.921356i
$$153$$ 0 0
$$154$$ −3000.00 −1.56978
$$155$$ 0 0
$$156$$ 0 0
$$157$$ − 1225.00i − 0.622711i −0.950293 0.311356i $$-0.899217\pi$$
0.950293 0.311356i $$-0.100783\pi$$
$$158$$ − 2770.16i − 1.39482i
$$159$$ 0 0
$$160$$ 0 0
$$161$$ −1707.63 −0.835901
$$162$$ 0 0
$$163$$ − 275.000i − 0.132145i −0.997815 0.0660726i $$-0.978953\pi$$
0.997815 0.0660726i $$-0.0210469\pi$$
$$164$$ 885.438 0.421592
$$165$$ 0 0
$$166$$ −1680.00 −0.785502
$$167$$ − 265.631i − 0.123085i −0.998104 0.0615424i $$-0.980398\pi$$
0.998104 0.0615424i $$-0.0196019\pi$$
$$168$$ 0 0
$$169$$ 972.000 0.442421
$$170$$ 0 0
$$171$$ 0 0
$$172$$ − 670.000i − 0.297018i
$$173$$ 3984.47i 1.75106i 0.483162 + 0.875531i $$0.339488\pi$$
−0.483162 + 0.875531i $$0.660512\pi$$
$$174$$ 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 $$0.612593\pi$$
$$182$$ − 1660.20i − 0.676164i
$$183$$ 0 0
$$184$$ −2160.00 −0.865420
$$185$$ 0 0
$$186$$ 0 0
$$187$$ − 5600.00i − 2.18991i
$$188$$ − 354.175i − 0.137398i
$$189$$ 0 0
$$190$$ 0 0
$$191$$ 2213.59 0.838587 0.419293 0.907851i $$-0.362278\pi$$
0.419293 + 0.907851i $$0.362278\pi$$
$$192$$ 0 0
$$193$$ 3625.00i 1.35199i 0.736908 + 0.675993i $$0.236285\pi$$
−0.736908 + 0.675993i $$0.763715\pi$$
$$194$$ 3431.07 1.26978
$$195$$ 0 0
$$196$$ −236.000 −0.0860058
$$197$$ 1745.58i 0.631306i 0.948875 + 0.315653i $$0.102224\pi$$
−0.948875 + 0.315653i $$0.897776\pi$$
$$198$$ 0 0
$$199$$ −3199.00 −1.13955 −0.569777 0.821800i $$-0.692970\pi$$
−0.569777 + 0.821800i $$0.692970\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ − 4200.00i − 1.46293i
$$203$$ − 948.683i − 0.328003i
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 4869.91 1.64710
$$207$$ 0 0
$$208$$ − 2660.00i − 0.886720i
$$209$$ −5755.35 −1.90481
$$210$$ 0 0
$$211$$ 887.000 0.289401 0.144700 0.989476i $$-0.453778\pi$$
0.144700 + 0.989476i $$0.453778\pi$$
$$212$$ − 177.088i − 0.0573699i
$$213$$ 0 0
$$214$$ −2520.00 −0.804970
$$215$$ 0 0
$$216$$ 0 0
$$217$$ 2205.00i 0.689793i
$$218$$ − 2564.61i − 0.796776i
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 3099.03 0.943274
$$222$$ 0 0
$$223$$ 3535.00i 1.06153i 0.847519 + 0.530765i $$0.178095\pi$$
−0.847519 + 0.530765i $$0.821905\pi$$
$$224$$ 1328.16 0.396166
$$225$$ 0 0
$$226$$ −5120.00 −1.50698
$$227$$ 88.5438i 0.0258892i 0.999916 + 0.0129446i $$0.00412052\pi$$
−0.999916 + 0.0129446i $$0.995879\pi$$
$$228$$ 0 0
$$229$$ 5019.00 1.44832 0.724159 0.689633i $$-0.242228\pi$$
0.724159 + 0.689633i $$0.242228\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ − 1200.00i − 0.339586i
$$233$$ − 3605.00i − 1.01361i −0.862061 0.506805i $$-0.830826\pi$$
0.862061 0.506805i $$-0.169174\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −1770.88 −0.488450
$$237$$ 0 0
$$238$$ 4200.00i 1.14389i
$$239$$ −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 $$0.526533\pi$$
$$242$$ − 8440.12i − 2.24195i
$$243$$ 0 0
$$244$$ −854.000 −0.224065
$$245$$ 0 0
$$246$$ 0 0
$$247$$ − 3185.00i − 0.820472i
$$248$$ 2789.13i 0.714153i
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 2656.31 0.667988 0.333994 0.942575i $$-0.391603\pi$$
0.333994 + 0.942575i $$0.391603\pi$$
$$252$$ 0 0
$$253$$ − 7200.00i − 1.78917i
$$254$$ −822.192 −0.203106
$$255$$ 0 0
$$256$$ 2896.00 0.707031
$$257$$ − 3718.84i − 0.902626i −0.892366 0.451313i $$-0.850956\pi$$
0.892366 0.451313i $$-0.149044\pi$$
$$258$$ 0 0
$$259$$ −5550.00 −1.33151
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 4200.00i 0.990370i
$$263$$ − 2896.65i − 0.679144i −0.940580 0.339572i $$-0.889718\pi$$
0.940580 0.339572i $$-0.110282\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 4316.51 0.994970
$$267$$ 0 0
$$268$$ 30.0000i 0.00683784i
$$269$$ −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 $$0.510990\pi$$
$$272$$ 6729.33i 1.50009i
$$273$$ 0 0
$$274$$ 8400.00 1.85205
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 7485.00i 1.62357i 0.583953 + 0.811787i $$0.301505\pi$$
−0.583953 + 0.811787i $$0.698495\pi$$
$$278$$ − 2479.23i − 0.534871i
$$279$$ 0 0
$$280$$ 0 0
$$281$$ −948.683 −0.201401 −0.100701 0.994917i $$-0.532108\pi$$
−0.100701 + 0.994917i $$0.532108\pi$$
$$282$$ 0 0
$$283$$ 525.000i 0.110276i 0.998479 + 0.0551378i $$0.0175598\pi$$
−0.998479 + 0.0551378i $$0.982440\pi$$
$$284$$ 126.491 0.0264291
$$285$$ 0 0
$$286$$ 7000.00 1.44727
$$287$$ 6640.78i 1.36583i
$$288$$ 0 0
$$289$$ −2927.00 −0.595766
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 140.000i 0.0280578i
$$293$$ 6375.15i 1.27113i 0.772048 + 0.635564i $$0.219232\pi$$
−0.772048 + 0.635564i $$0.780768\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −7020.26 −1.37853
$$297$$ 0 0
$$298$$ − 6800.00i − 1.32186i
$$299$$ 3984.47 0.770662
$$300$$ 0 0
$$301$$ 5025.00 0.962246
$$302$$ 2520.34i 0.480228i
$$303$$ 0 0
$$304$$ 6916.00 1.30480
$$305$$ 0 0
$$306$$ 0 0
$$307$$ 2695.00i 0.501016i 0.968115 + 0.250508i $$0.0805976\pi$$
−0.968115 + 0.250508i $$0.919402\pi$$
$$308$$ 1897.37i 0.351015i
$$309$$ 0 0
$$310$$ 0 0
$$311$$ −5312.63 −0.968654 −0.484327 0.874887i $$-0.660936\pi$$
−0.484327 + 0.874887i $$0.660936\pi$$
$$312$$ 0 0
$$313$$ − 2555.00i − 0.461397i −0.973025 0.230698i $$-0.925899\pi$$
0.973025 0.230698i $$-0.0741010\pi$$
$$314$$ −3873.79 −0.696212
$$315$$ 0 0
$$316$$ −1752.00 −0.311891
$$317$$ − 7462.98i − 1.32228i −0.750263 0.661140i $$-0.770073\pi$$
0.750263 0.661140i $$-0.229927\pi$$
$$318$$ 0 0
$$319$$ 4000.00 0.702060
$$320$$ 0 0
$$321$$ 0 0
$$322$$ 5400.00i 0.934566i
$$323$$ 8057.48i 1.38802i
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −869.626 −0.147743
$$327$$ 0 0
$$328$$ 8400.00i 1.41406i
$$329$$ 2656.31 0.445129
$$330$$ 0 0
$$331$$ −5088.00 −0.844900 −0.422450 0.906386i $$-0.638830\pi$$
−0.422450 + 0.906386i $$0.638830\pi$$
$$332$$ 1062.53i 0.175644i
$$333$$ 0 0
$$334$$ −840.000 −0.137613
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 11485.0i 1.85646i 0.372004 + 0.928231i $$0.378671\pi$$
−0.372004 + 0.928231i $$0.621329\pi$$
$$338$$ − 3073.73i − 0.494642i
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −9297.10 −1.47644
$$342$$ 0 0
$$343$$ − 6915.00i − 1.08856i
$$344$$ 6356.18 0.996227
$$345$$ 0 0
$$346$$ 12600.0 1.95775
$$347$$ 6400.45i 0.990185i 0.868840 + 0.495092i $$0.164866\pi$$
−0.868840 + 0.495092i $$0.835134\pi$$
$$348$$ 0 0
$$349$$ 9674.00 1.48377 0.741887 0.670525i $$-0.233931\pi$$
0.741887 + 0.670525i $$0.233931\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 5600.00i 0.847957i
$$353$$ − 5578.26i − 0.841078i −0.907274 0.420539i $$-0.861841\pi$$
0.907274 0.420539i $$-0.138159\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ 0 0
$$357$$ 0 0
$$358$$ 5600.00i 0.826730i
$$359$$ 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
$$362$$ 5334.76i 0.774555i
$$363$$ 0 0
$$364$$ −1050.00 −0.151195
$$365$$ 0 0
$$366$$ 0 0
$$367$$ − 8505.00i − 1.20969i −0.796342 0.604847i $$-0.793234\pi$$
0.796342 0.604847i $$-0.206766\pi$$
$$368$$ 8651.99i 1.22559i
$$369$$ 0 0
$$370$$ 0 0
$$371$$ 1328.16 0.185861
$$372$$ 0 0
$$373$$ − 6775.00i − 0.940472i −0.882541 0.470236i $$-0.844169\pi$$
0.882541 0.470236i $$-0.155831\pi$$
$$374$$ −17708.8 −2.44839
$$375$$ 0 0
$$376$$ 3360.00 0.460848
$$377$$ 2213.59i 0.302403i
$$378$$ 0 0
$$379$$ −9241.00 −1.25245 −0.626225 0.779643i $$-0.715401\pi$$
−0.626225 + 0.779643i $$0.715401\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ − 7000.00i − 0.937568i
$$383$$ − 9031.46i − 1.20493i −0.798147 0.602463i $$-0.794186\pi$$
0.798147 0.602463i $$-0.205814\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 11463.3 1.51157
$$387$$ 0 0
$$388$$ − 2170.00i − 0.283931i
$$389$$ 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
$$392$$ − 2238.89i − 0.288472i
$$393$$ 0 0
$$394$$ 5520.00 0.705821
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 4795.00i 0.606182i 0.952962 + 0.303091i $$0.0980186\pi$$
−0.952962 + 0.303091i $$0.901981\pi$$
$$398$$ 10116.1i 1.27406i
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 7020.26 0.874252 0.437126 0.899400i $$-0.355996\pi$$
0.437126 + 0.899400i $$0.355996\pi$$
$$402$$ 0 0
$$403$$ − 5145.00i − 0.635957i
$$404$$ −2656.31 −0.327120
$$405$$ 0 0
$$406$$ −3000.00 −0.366718
$$407$$ − 23400.9i − 2.84997i
$$408$$ 0 0
$$409$$ −6881.00 −0.831891 −0.415946 0.909389i $$-0.636549\pi$$
−0.415946 + 0.909389i $$0.636549\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ − 3080.00i − 0.368303i
$$413$$ − 13281.6i − 1.58243i
$$414$$ 0 0
$$415$$ 0 0
$$416$$ −3099.03 −0.365247
$$417$$ 0 0
$$418$$ 18200.0i 2.12964i
$$419$$ −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 $$0.378847\pi$$
$$422$$ − 2804.94i − 0.323560i
$$423$$ 0 0
$$424$$ 1680.00 0.192425
$$425$$ 0 0
$$426$$ 0 0
$$427$$ − 6405.00i − 0.725901i
$$428$$ 1593.79i 0.179997i
$$429$$ 0 0
$$430$$ 0 0
$$431$$ 1328.16 0.148434 0.0742170 0.997242i $$-0.476354\pi$$
0.0742170 + 0.997242i $$0.476354\pi$$
$$432$$ 0 0
$$433$$ − 8155.00i − 0.905091i −0.891741 0.452545i $$-0.850516\pi$$
0.891741 0.452545i $$-0.149484\pi$$
$$434$$ 6972.82 0.771212
$$435$$ 0 0
$$436$$ −1622.00 −0.178164
$$437$$ 10359.6i 1.13402i
$$438$$ 0 0
$$439$$ −14749.0 −1.60349 −0.801744 0.597667i $$-0.796094\pi$$
−0.801744 + 0.597667i $$0.796094\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ − 9800.00i − 1.05461i
$$443$$ − 11156.5i − 1.19653i −0.801299 0.598264i $$-0.795857\pi$$
0.801299 0.598264i $$-0.204143\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ 11178.7 1.18683
$$447$$ 0 0
$$448$$ 4920.00i 0.518857i
$$449$$ 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
$$452$$ 3238.17i 0.336971i
$$453$$ 0 0
$$454$$ 280.000 0.0289450
$$455$$ 0 0
$$456$$ 0 0
$$457$$ 1030.00i 0.105430i 0.998610 + 0.0527148i $$0.0167874\pi$$
−0.998610 + 0.0527148i $$0.983213\pi$$
$$458$$ − 15871.5i − 1.61927i
$$459$$ 0 0
$$460$$ 0 0
$$461$$ 9297.10 0.939282 0.469641 0.882858i $$-0.344383\pi$$
0.469641 + 0.882858i $$0.344383\pi$$
$$462$$ 0 0
$$463$$ 11940.0i 1.19849i 0.800567 + 0.599243i $$0.204532\pi$$
−0.800567 + 0.599243i $$0.795468\pi$$
$$464$$ −4806.66 −0.480913
$$465$$ 0 0
$$466$$ −11400.0 −1.13325
$$467$$ − 12130.5i − 1.20200i −0.799250 0.600998i $$-0.794770\pi$$
0.799250 0.600998i $$-0.205230\pi$$
$$468$$ 0 0
$$469$$ −225.000 −0.0221525
$$470$$ 0 0
$$471$$ 0 0
$$472$$ − 16800.0i − 1.63831i
$$473$$ 21187.3i 2.05960i
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 2656.31 0.255781
$$477$$ 0 0
$$478$$ 1400.00i 0.133963i
$$479$$ 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
$$482$$ 1970.10i 0.186173i
$$483$$ 0 0
$$484$$ −5338.00 −0.501315
$$485$$ 0 0
$$486$$ 0 0
$$487$$ − 14125.0i − 1.31430i −0.753759 0.657151i $$-0.771761\pi$$
0.753759 0.657151i $$-0.228239\pi$$
$$488$$ − 8101.76i − 0.751535i
$$489$$ 0 0
$$490$$ 0 0
$$491$$ −4047.72 −0.372038 −0.186019 0.982546i $$-0.559559\pi$$
−0.186019 + 0.982546i $$0.559559\pi$$
$$492$$ 0 0
$$493$$ − 5600.00i − 0.511585i
$$494$$ −10071.9 −0.917316
$$495$$ 0 0
$$496$$ 11172.0 1.01137
$$497$$ 948.683i 0.0856223i
$$498$$ 0 0
$$499$$ −1279.00 −0.114741 −0.0573706 0.998353i $$-0.518272\pi$$
−0.0573706 + 0.998353i $$0.518272\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ − 8400.00i − 0.746833i
$$503$$ 16380.6i 1.45204i 0.687675 + 0.726019i $$0.258631\pi$$
−0.687675 + 0.726019i $$0.741369\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −22768.4 −2.00035
$$507$$ 0 0
$$508$$ 520.000i 0.0454159i
$$509$$ −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
$$512$$ 4806.66i 0.414895i
$$513$$ 0 0
$$514$$ −11760.0 −1.00917
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 11200.0i 0.952757i
$$518$$ 17550.6i 1.48867i
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −11068.0 −0.930704 −0.465352 0.885126i $$-0.654072\pi$$
−0.465352 + 0.885126i $$0.654072\pi$$
$$522$$ 0 0
$$523$$ 10745.0i 0.898367i 0.893439 + 0.449184i $$0.148285\pi$$
−0.893439 + 0.449184i $$0.851715\pi$$
$$524$$ 2656.31 0.221453
$$525$$ 0 0
$$526$$ −9160.00 −0.759306
$$527$$ 13015.9i 1.07587i
$$528$$ 0 0
$$529$$ −793.000 −0.0651763
$$530$$ 0 0
$$531$$ 0 0
$$532$$ − 2730.00i − 0.222482i
$$533$$ − 15495.2i − 1.25923i
$$534$$ 0 0
$$535$$ 0 0
$$536$$ −284.605 −0.0229348
$$537$$ 0 0
$$538$$ 1400.00i 0.112190i
$$539$$ 7462.98 0.596388
$$540$$ 0 0
$$541$$ −9423.00 −0.748847 −0.374424 0.927258i $$-0.622159\pi$$
−0.374424 + 0.927258i $$0.622159\pi$$
$$542$$ 973.982i 0.0771884i
$$543$$ 0 0
$$544$$ 7840.00 0.617899
$$545$$ 0 0
$$546$$ 0 0
$$547$$ − 9080.00i − 0.709749i −0.934914 0.354875i $$-0.884524\pi$$
0.934914 0.354875i $$-0.115476\pi$$
$$548$$ − 5312.63i − 0.414132i
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −5755.35 −0.444984
$$552$$ 0 0
$$553$$ − 13140.0i − 1.01043i
$$554$$ 23669.6 1.81521
$$555$$ 0 0
$$556$$ −1568.00 −0.119601
$$557$$ 8348.41i 0.635069i 0.948247 + 0.317535i $$0.102855\pi$$
−0.948247 + 0.317535i $$0.897145\pi$$
$$558$$ 0 0
$$559$$ −11725.0 −0.887146
$$560$$ 0 0
$$561$$ 0 0
$$562$$ 3000.00i 0.225173i
$$563$$ 7703.31i 0.576653i 0.957532 + 0.288327i $$0.0930989\pi$$
−0.957532 + 0.288327i $$0.906901\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ 1660.20 0.123292
$$567$$ 0 0
$$568$$ 1200.00i 0.0886459i
$$569$$ 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 $$0.493152\pi$$
$$572$$ − 4427.19i − 0.323619i
$$573$$ 0 0
$$574$$ 21000.0 1.52704
$$575$$ 0 0
$$576$$ 0 0
$$577$$ 15995.0i 1.15404i 0.816730 + 0.577020i $$0.195784\pi$$
−0.816730 + 0.577020i $$0.804216\pi$$
$$578$$ 9255.99i 0.666087i
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −7968.94 −0.569032
$$582$$ 0 0
$$583$$ 5600.00i 0.397819i
$$584$$ −1328.16 −0.0941088
$$585$$ 0 0
$$586$$ 20160.0 1.42116
$$587$$ 20453.6i 1.43818i 0.694918 + 0.719089i $$0.255441\pi$$
−0.694918 + 0.719089i $$0.744559\pi$$
$$588$$ 0 0
$$589$$ 13377.0 0.935806
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 28120.0i 1.95224i
$$593$$ 11599.2i 0.803244i 0.915806 + 0.401622i $$0.131553\pi$$
−0.915806 + 0.401622i $$0.868447\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −4300.70 −0.295576
$$597$$ 0 0
$$598$$ − 12600.0i − 0.861626i
$$599$$ −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 $$0.465984\pi$$
$$602$$ − 15890.4i − 1.07582i
$$603$$ 0 0
$$604$$ 1594.00 0.107382
$$605$$ 0 0
$$606$$ 0 0
$$607$$ − 2660.00i − 0.177868i −0.996038 0.0889342i $$-0.971654\pi$$
0.996038 0.0889342i $$-0.0283461\pi$$
$$608$$ − 8057.48i − 0.537457i
$$609$$ 0 0
$$610$$ 0 0
$$611$$ −6198.06 −0.410388
$$612$$ 0 0
$$613$$ − 6670.00i − 0.439476i −0.975559 0.219738i $$-0.929480\pi$$
0.975559 0.219738i $$-0.0705202\pi$$
$$614$$ 8522.34 0.560152
$$615$$ 0 0
$$616$$ −18000.0 −1.17734
$$617$$ − 11042.7i − 0.720521i −0.932852 0.360260i $$-0.882688\pi$$
0.932852 0.360260i $$-0.117312\pi$$
$$618$$ 0 0
$$619$$ −5579.00 −0.362260 −0.181130 0.983459i $$-0.557975\pi$$
−0.181130 + 0.983459i $$0.557975\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ 16800.0i 1.08299i
$$623$$ 0 0
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −8079.62 −0.515857
$$627$$ 0 0
$$628$$ 2450.00i 0.155678i
$$629$$ −32761.2 −2.07675
$$630$$ 0 0
$$631$$ 23617.0 1.48998 0.744990 0.667075i $$-0.232454\pi$$
0.744990 + 0.667075i $$0.232454\pi$$
$$632$$ − 16620.9i − 1.04612i
$$633$$ 0 0
$$634$$ −23600.0 −1.47835
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 4130.00i 0.256886i
$$638$$ − 12649.1i − 0.784926i
$$639$$ 0 0
$$640$$ 0 0
$$641$$ 31496.3 1.94076 0.970381 0.241579i $$-0.0776655\pi$$
0.970381 + 0.241579i $$0.0776655\pi$$
$$642$$ 0 0
$$643$$ − 14280.0i − 0.875814i −0.899020 0.437907i $$-0.855720\pi$$
0.899020 0.437907i $$-0.144280\pi$$
$$644$$ 3415.26 0.208975
$$645$$ 0 0
$$646$$ 25480.0 1.55185
$$647$$ 12396.1i 0.753234i 0.926369 + 0.376617i $$0.122913\pi$$
−0.926369 + 0.376617i $$0.877087\pi$$
$$648$$ 0 0
$$649$$ 56000.0 3.38705
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 550.000i 0.0330363i
$$653$$ − 12042.0i − 0.721651i −0.932633 0.360825i $$-0.882495\pi$$
0.932633 0.360825i $$-0.117505\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ 33646.6 2.00256
$$657$$ 0 0
$$658$$ − 8400.00i − 0.497669i
$$659$$ −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 $$0.230808\pi$$
$$662$$ 16089.7i 0.944626i
$$663$$ 0 0
$$664$$ −10080.0 −0.589126
$$665$$ 0 0
$$666$$ 0 0
$$667$$ − 7200.00i − 0.417969i
$$668$$ 531.263i 0.0307712i
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 27005.9 1.55372
$$672$$ 0 0
$$673$$ 15150.0i 0.867741i 0.900975 + 0.433870i $$0.142852\pi$$
−0.900975 + 0.433870i $$0.857148\pi$$
$$674$$ 36318.8 2.07559
$$675$$ 0 0
$$676$$ −1944.00 −0.110605
$$677$$ − 20984.9i − 1.19131i −0.803242 0.595653i $$-0.796893\pi$$
0.803242 0.595653i $$-0.203107\pi$$
$$678$$ 0 0
$$679$$ 16275.0 0.919849
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 29400.0i 1.65071i
$$683$$ − 1973.26i − 0.110549i −0.998471 0.0552743i $$-0.982397\pi$$
0.998471 0.0552743i $$-0.0176033\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −21867.2 −1.21704
$$687$$ 0 0
$$688$$ − 25460.0i − 1.41083i
$$689$$ −3099.03 −0.171355
$$690$$ 0 0
$$691$$ −20552.0 −1.13145 −0.565727 0.824592i $$-0.691404\pi$$
−0.565727 + 0.824592i $$0.691404\pi$$
$$692$$ − 7968.94i − 0.437765i
$$693$$ 0 0
$$694$$ 20240.0 1.10706
$$695$$ 0 0
$$696$$ 0 0
$$697$$ 39200.0i 2.13028i
$$698$$ − 30591.9i − 1.65891i
$$699$$ 0 0
$$700$$ 0 0
$$701$$ −8411.66 −0.453215 −0.226608 0.973986i $$-0.572764\pi$$
−0.226608 + 0.973986i $$0.572764\pi$$
$$702$$ 0 0
$$703$$ 33670.0i 1.80638i
$$704$$ −20744.5 −1.11057
$$705$$ 0 0
$$706$$ −17640.0 −0.940354
$$707$$ − 19922.3i − 1.05977i
$$708$$ 0 0
$$709$$ −17281.0 −0.915376 −0.457688 0.889113i $$-0.651322\pi$$
−0.457688 + 0.889113i $$0.651322\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 0 0
$$713$$ 16734.8i 0.878993i
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 3541.75 0.184862
$$717$$ 0 0
$$718$$ − 28200.0i − 1.46576i
$$719$$ 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
$$722$$ − 4496.76i − 0.231790i
$$723$$ 0 0
$$724$$ 3374.00 0.173196
$$725$$ 0 0
$$726$$ 0 0
$$727$$ − 30415.0i − 1.55162i −0.630965 0.775811i $$-0.717341\pi$$
0.630965 0.775811i $$-0.282659\pi$$
$$728$$ − 9961.17i − 0.507123i
$$729$$ 0 0
$$730$$ 0 0
$$731$$ 29662.2 1.50081
$$732$$ 0 0
$$733$$ 18550.0i 0.934734i 0.884063 + 0.467367i $$0.154797\pi$$
−0.884063 + 0.467367i $$0.845203\pi$$
$$734$$ −26895.2 −1.35248
$$735$$ 0 0
$$736$$ 10080.0 0.504828
$$737$$ − 948.683i − 0.0474155i
$$738$$ 0 0
$$739$$ −24016.0 −1.19546 −0.597729 0.801699i $$-0.703930\pi$$
−0.597729 + 0.801699i $$0.703930\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ − 4200.00i − 0.207799i
$$743$$ − 12396.1i − 0.612072i −0.952020 0.306036i $$-0.900997\pi$$
0.952020 0.306036i $$-0.0990029\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ −21424.4 −1.05148
$$747$$ 0 0
$$748$$ 11200.0i 0.547477i
$$749$$ −11953.4 −0.583135
$$750$$ 0 0
$$751$$ −7772.00 −0.377636 −0.188818 0.982012i $$-0.560466\pi$$
−0.188818 + 0.982012i $$0.560466\pi$$
$$752$$ − 13458.7i − 0.652642i
$$753$$ 0 0
$$754$$ 7000.00 0.338097
$$755$$ 0 0
$$756$$ 0 0
$$757$$ 3865.00i 0.185569i 0.995686 + 0.0927846i $$0.0295768\pi$$
−0.995686 + 0.0927846i $$0.970423\pi$$
$$758$$ 29222.6i 1.40028i
$$759$$ 0 0
$$760$$ 0 0
$$761$$ −15937.9 −0.759195 −0.379598 0.925152i $$-0.623938\pi$$
−0.379598 + 0.925152i $$0.623938\pi$$
$$762$$ 0 0
$$763$$ − 12165.0i − 0.577199i
$$764$$ −4427.19 −0.209647
$$765$$ 0 0
$$766$$ −28560.0 −1.34715
$$767$$ 30990.3i 1.45893i
$$768$$ 0 0
$$769$$ 7441.00 0.348933 0.174466 0.984663i $$-0.444180\pi$$
0.174466 + 0.984663i $$0.444180\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ − 7250.00i − 0.337996i
$$773$$ 25766.2i 1.19890i 0.800413 + 0.599448i $$0.204613\pi$$
−0.800413 + 0.599448i $$0.795387\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ 20586.4 0.952332
$$777$$ 0 0
$$778$$ − 13800.0i − 0.635931i
$$779$$ 40287.4 1.85295
$$780$$ 0 0
$$781$$ −4000.00 −0.183267
$$782$$ 31875.8i 1.45764i
$$783$$ 0 0
$$784$$ −8968.00 −0.408528
$$785$$ 0 0
$$786$$ 0 0
$$787$$ − 6125.00i − 0.277424i −0.990333 0.138712i $$-0.955704\pi$$
0.990333 0.138712i $$-0.0442962\pi$$
$$788$$ − 3491.15i − 0.157826i
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −24286.3 −1.09168
$$792$$ 0 0
$$793$$ 14945.0i 0.669247i
$$794$$ 15163.1 0.677732
$$795$$ 0 0
$$796$$ 6398.00 0.284888
$$797$$ 20010.9i 0.889363i 0.895689 + 0.444681i $$0.146683\pi$$
−0.895689 + 0.444681i $$0.853317\pi$$
$$798$$ 0 0
$$799$$ 15680.0 0.694266
$$800$$ 0 0
$$801$$ 0 0
$$802$$ − 22200.0i − 0.977443i
$$803$$ − 4427.19i − 0.194561i
$$804$$ 0 0
$$805$$ 0 0
$$806$$ −16269.9 −0.711022
$$807$$ 0 0
$$808$$ − 25200.0i − 1.09719i
$$809$$ 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 $$0.287270\pi$$
$$812$$ 1897.37i 0.0820006i
$$813$$ 0 0
$$814$$ −74000.0 −3.18636
$$815$$ 0 0
$$816$$ 0 0
$$817$$ − 30485.0i − 1.30543i
$$818$$ 21759.6i 0.930083i
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −26563.1 −1.12918 −0.564592 0.825370i $$-0.690966\pi$$
−0.564592 + 0.825370i $$0.690966\pi$$
$$822$$ 0 0
$$823$$ 1135.00i 0.0480724i 0.999711 + 0.0240362i $$0.00765170\pi$$
−0.999711 + 0.0240362i $$0.992348\pi$$
$$824$$ 29219.4 1.23532
$$825$$ 0 0
$$826$$ −42000.0 −1.76921
$$827$$ − 999.280i − 0.0420174i −0.999779 0.0210087i $$-0.993312\pi$$
0.999779 0.0210087i $$-0.00668776\pi$$
$$828$$ 0 0
$$829$$ 1974.00 0.0827019 0.0413509 0.999145i $$-0.486834\pi$$
0.0413509 + 0.999145i $$0.486834\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ − 11480.0i − 0.478362i
$$833$$ − 10448.2i − 0.434583i
$$834$$ 0 0
$$835$$ 0 0
$$836$$ 11510.7 0.476203
$$837$$ 0 0
$$838$$ 40600.0i 1.67363i
$$839$$ −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
$$842$$ − 20295.5i − 0.830676i
$$843$$ 0 0
$$844$$ −1774.00 −0.0723502
$$845$$ 0 0
$$846$$ 0 0
$$847$$ − 40035.0i − 1.62411i
$$848$$ − 6729.33i − 0.272507i
$$849$$ 0 0
$$850$$ 0 0
$$851$$ −42121.5 −1.69672
$$852$$ 0 0
$$853$$ 36645.0i 1.47093i 0.677564 + 0.735464i $$0.263036\pi$$
−0.677564 + 0.735464i $$0.736964\pi$$
$$854$$ −20254.4 −0.811582
$$855$$ 0 0
$$856$$ −15120.0 −0.603728
$$857$$ 10802.3i 0.430573i 0.976551 + 0.215286i $$0.0690685\pi$$
−0.976551 + 0.215286i $$0.930931\pi$$
$$858$$ 0 0
$$859$$ −20104.0 −0.798533 −0.399266 0.916835i $$-0.630735\pi$$
−0.399266 + 0.916835i $$0.630735\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ − 4200.00i − 0.165954i
$$863$$ 2150.35i 0.0848189i 0.999100 + 0.0424095i $$0.0135034\pi$$
−0.999100 + 0.0424095i $$0.986497\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −25788.4 −1.01192
$$867$$ 0 0
$$868$$ − 4410.00i − 0.172448i
$$869$$ 55403.1 2.16274
$$870$$ 0 0
$$871$$ 525.000 0.0204236
$$872$$ − 15387.6i − 0.597582i
$$873$$ 0 0
$$874$$ 32760.0 1.26788
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 21975.0i 0.846115i 0.906103 + 0.423058i $$0.139043\pi$$
−0.906103 + 0.423058i $$0.860957\pi$$
$$878$$ 46640.4i 1.79275i
$$879$$ 0 0
$$880$$ 0 0
$$881$$ −885.438 −0.0338606 −0.0169303 0.999857i $$-0.505389\pi$$
−0.0169303 + 0.999857i $$0.505389\pi$$
$$882$$ 0 0
$$883$$ 34915.0i 1.33067i 0.746544 + 0.665336i $$0.231712\pi$$
−0.746544 + 0.665336i $$0.768288\pi$$
$$884$$ −6198.06 −0.235818
$$885$$ 0 0
$$886$$ −35280.0 −1.33776
$$887$$ − 49407.4i − 1.87028i −0.354277 0.935140i $$-0.615273\pi$$
0.354277 0.935140i $$-0.384727\pi$$
$$888$$ 0 0
$$889$$ −3900.00 −0.147134
$$890$$ 0 0
$$891$$ 0 0
$$892$$ − 7070.00i − 0.265382i
$$893$$ − 16115.0i − 0.603882i
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 26183.7 0.976266
$$897$$ 0 0
$$898$$ − 29200.0i − 1.08510i
$$899$$ −9297.10 −0.344912
$$900$$ 0 0
$$901$$ 7840.00 0.289887
$$902$$ 88543.8i 3.26850i
$$903$$ 0 0
$$904$$ −30720.0 −1.13023
$$905$$ 0 0
$$906$$ 0 0
$$907$$ − 17880.0i − 0.654571i −0.944926 0.327285i $$-0.893866\pi$$
0.944926 0.327285i $$-0.106134\pi$$
$$908$$ − 177.088i − 0.00647231i
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 17329.3 0.630236 0.315118 0.949053i $$-0.397956\pi$$
0.315118 + 0.949053i $$0.397956\pi$$
$$912$$ 0 0
$$913$$ − 33600.0i − 1.21796i
$$914$$ 3257.15 0.117874
$$915$$ 0 0
$$916$$ −10038.0 −0.362080
$$917$$ 19922.3i 0.717442i
$$918$$ 0 0
$$919$$ 25829.0 0.927117 0.463558 0.886066i $$-0.346572\pi$$
0.463558 + 0.886066i $$0.346572\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ − 29400.0i − 1.05015i
$$923$$ − 2213.59i − 0.0789397i
$$924$$ 0 0
$$925$$ 0 0
$$926$$ 37757.6 1.33995
$$927$$ 0 0
$$928$$ 5600.00i 0.198092i
$$929$$ 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
$$932$$ 7209.99i 0.253403i
$$933$$ 0 0
$$934$$ −38360.0 −1.34387
$$935$$ 0 0
$$936$$ 0 0
$$937$$ − 40005.0i − 1.39478i −0.716693 0.697389i $$-0.754345\pi$$
0.716693 0.697389i $$-0.245655\pi$$
$$938$$ 711.512i 0.0247673i
$$939$$ 0 0
$$940$$ 0 0
$$941$$ 30104.9 1.04292 0.521462 0.853275i $$-0.325387\pi$$
0.521462 + 0.853275i $$0.325387\pi$$
$$942$$ 0 0
$$943$$ 50400.0i 1.74046i
$$944$$ −67293.3 −2.32014
$$945$$ 0 0
$$946$$ 67000.0 2.30270
$$947$$ − 43297.9i − 1.48574i −0.669437 0.742868i $$-0.733465\pi$$
0.669437 0.742868i $$-0.266535\pi$$
$$948$$ 0 0
$$949$$ 2450.00 0.0838044
$$950$$ 0 0
$$951$$ 0 0
$$952$$ 25200.0i 0.857917i
$$953$$ − 53682.8i − 1.82472i −0.409390 0.912360i $$-0.634258\pi$$
0.409390 0.912360i $$-0.365742\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 885.438 0.0299551
$$957$$ 0 0
$$958$$ − 8400.00i − 0.283290i
$$959$$ 39844.7 1.34166
$$960$$ 0 0
$$961$$ −8182.00 −0.274647
$$962$$ − 40951.5i − 1.37248i
$$963$$ 0 0
$$964$$ 1246.00 0.0416296
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 20540.0i 0.683063i 0.939870 + 0.341531i $$0.110946\pi$$
−0.939870 + 0.341531i $$0.889054\pi$$
$$968$$ − 50640.7i − 1.68146i
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 10625.3 0.351164 0.175582 0.984465i $$-0.443819\pi$$
0.175582 + 0.984465i $$0.443819\pi$$
$$972$$ 0 0
$$973$$ − 11760.0i − 0.387470i
$$974$$ −44667.2 −1.46943
$$975$$ 0 0
$$976$$ −32452.0 −1.06431
$$977$$ − 28359.3i − 0.928654i −0.885664 0.464327i $$-0.846296\pi$$
0.885664 0.464327i $$-0.153704\pi$$
$$978$$ 0 0
$$979$$ 0 0
$$980$$ 0 0
$$981$$ 0 0
$$982$$ 12800.0i 0.415952i
$$983$$ 35063.3i 1.13769i 0.822446 + 0.568844i $$0.192609\pi$$
−0.822446 + 0.568844i $$0.807391\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −17708.8 −0.571969
$$987$$ 0 0
$$988$$ 6370.00i 0.205118i
$$989$$ 38137.1 1.22618
$$990$$ 0 0
$$991$$ −18283.0 −0.586053 −0.293027 0.956104i $$-0.594662\pi$$
−0.293027 + 0.956104i $$0.594662\pi$$
$$992$$ − 13015.9i − 0.416589i
$$993$$ 0 0
$$994$$ 3000.00 0.0957286
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 13230.0i 0.420259i 0.977674 + 0.210130i $$0.0673886\pi$$
−0.977674 + 0.210130i $$0.932611\pi$$
$$998$$ 4044.55i 0.128285i
$$999$$ 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.b.i.199.1 4
3.2 odd 2 inner 225.4.b.i.199.3 4
5.2 odd 4 225.4.a.m.1.2 yes 2
5.3 odd 4 225.4.a.l.1.1 2
5.4 even 2 inner 225.4.b.i.199.4 4
15.2 even 4 225.4.a.m.1.1 yes 2
15.8 even 4 225.4.a.l.1.2 yes 2
15.14 odd 2 inner 225.4.b.i.199.2 4

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