# Properties

 Label 2475.2.a.o.1.1 Level $2475$ Weight $2$ Character 2475.1 Self dual yes Analytic conductor $19.763$ Analytic rank $0$ Dimension $2$ CM no Inner twists $1$

# Related objects

## Newspace parameters

 Level: $$N$$ $$=$$ $$2475 = 3^{2} \cdot 5^{2} \cdot 11$$ Weight: $$k$$ $$=$$ $$2$$ Character orbit: $$[\chi]$$ $$=$$ 2475.a (trivial)

## Newform invariants

 Self dual: yes Analytic conductor: $$19.7629745003$$ Analytic rank: $$0$$ Dimension: $$2$$ Coefficient field: $$\Q(\sqrt{13})$$ Defining polynomial: $$x^{2} - x - 3$$ Coefficient ring: $$\Z[a_1, a_2]$$ Coefficient ring index: $$1$$ Twist minimal: no (minimal twist has level 275) Fricke sign: $$-1$$ Sato-Tate group: $\mathrm{SU}(2)$

## Embedding invariants

 Embedding label 1.1 Root $$2.30278$$ of defining polynomial Character $$\chi$$ $$=$$ 2475.1

## $q$-expansion

 $$f(q)$$ $$=$$ $$q-2.30278 q^{2} +3.30278 q^{4} +4.30278 q^{7} -3.00000 q^{8} +O(q^{10})$$ $$q-2.30278 q^{2} +3.30278 q^{4} +4.30278 q^{7} -3.00000 q^{8} +1.00000 q^{11} +5.00000 q^{13} -9.90833 q^{14} +0.302776 q^{16} +3.90833 q^{17} -1.00000 q^{19} -2.30278 q^{22} +3.69722 q^{23} -11.5139 q^{26} +14.2111 q^{28} +9.90833 q^{29} -4.21110 q^{31} +5.30278 q^{32} -9.00000 q^{34} +9.60555 q^{37} +2.30278 q^{38} -1.60555 q^{41} -7.21110 q^{43} +3.30278 q^{44} -8.51388 q^{46} +3.00000 q^{47} +11.5139 q^{49} +16.5139 q^{52} -2.30278 q^{53} -12.9083 q^{56} -22.8167 q^{58} -0.211103 q^{59} +2.90833 q^{61} +9.69722 q^{62} -12.8167 q^{64} -4.00000 q^{67} +12.9083 q^{68} -4.60555 q^{71} +2.90833 q^{73} -22.1194 q^{74} -3.30278 q^{76} +4.30278 q^{77} -0.0916731 q^{79} +3.69722 q^{82} -14.5139 q^{83} +16.6056 q^{86} -3.00000 q^{88} -5.30278 q^{89} +21.5139 q^{91} +12.2111 q^{92} -6.90833 q^{94} +11.6972 q^{97} -26.5139 q^{98} +O(q^{100})$$ $$\operatorname{Tr}(f)(q)$$ $$=$$ $$2 q - q^{2} + 3 q^{4} + 5 q^{7} - 6 q^{8} + O(q^{10})$$ $$2 q - q^{2} + 3 q^{4} + 5 q^{7} - 6 q^{8} + 2 q^{11} + 10 q^{13} - 9 q^{14} - 3 q^{16} - 3 q^{17} - 2 q^{19} - q^{22} + 11 q^{23} - 5 q^{26} + 14 q^{28} + 9 q^{29} + 6 q^{31} + 7 q^{32} - 18 q^{34} + 12 q^{37} + q^{38} + 4 q^{41} + 3 q^{44} + q^{46} + 6 q^{47} + 5 q^{49} + 15 q^{52} - q^{53} - 15 q^{56} - 24 q^{58} + 14 q^{59} - 5 q^{61} + 23 q^{62} - 4 q^{64} - 8 q^{67} + 15 q^{68} - 2 q^{71} - 5 q^{73} - 19 q^{74} - 3 q^{76} + 5 q^{77} - 11 q^{79} + 11 q^{82} - 11 q^{83} + 26 q^{86} - 6 q^{88} - 7 q^{89} + 25 q^{91} + 10 q^{92} - 3 q^{94} + 27 q^{97} - 35 q^{98} + O(q^{100})$$

## 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
$$n$$ $$a_n$$ $$a_n / n^{(k-1)/2}$$ $$\alpha_n$$ $$\theta_n$$
$$p$$ $$a_p$$ $$a_p / p^{(k-1)/2}$$ $$\alpha_p$$ $$\theta_p$$
$$2$$ −2.30278 −1.62831 −0.814154 0.580649i $$-0.802799\pi$$
−0.814154 + 0.580649i $$0.802799\pi$$
$$3$$ 0 0
$$4$$ 3.30278 1.65139
$$5$$ 0 0
$$6$$ 0 0
$$7$$ 4.30278 1.62630 0.813148 0.582057i $$-0.197752\pi$$
0.813148 + 0.582057i $$0.197752\pi$$
$$8$$ −3.00000 −1.06066
$$9$$ 0 0
$$10$$ 0 0
$$11$$ 1.00000 0.301511
$$12$$ 0 0
$$13$$ 5.00000 1.38675 0.693375 0.720577i $$-0.256123\pi$$
0.693375 + 0.720577i $$0.256123\pi$$
$$14$$ −9.90833 −2.64811
$$15$$ 0 0
$$16$$ 0.302776 0.0756939
$$17$$ 3.90833 0.947909 0.473954 0.880549i $$-0.342826\pi$$
0.473954 + 0.880549i $$0.342826\pi$$
$$18$$ 0 0
$$19$$ −1.00000 −0.229416 −0.114708 0.993399i $$-0.536593\pi$$
−0.114708 + 0.993399i $$0.536593\pi$$
$$20$$ 0 0
$$21$$ 0 0
$$22$$ −2.30278 −0.490953
$$23$$ 3.69722 0.770925 0.385462 0.922724i $$-0.374042\pi$$
0.385462 + 0.922724i $$0.374042\pi$$
$$24$$ 0 0
$$25$$ 0 0
$$26$$ −11.5139 −2.25806
$$27$$ 0 0
$$28$$ 14.2111 2.68565
$$29$$ 9.90833 1.83993 0.919965 0.392000i $$-0.128217\pi$$
0.919965 + 0.392000i $$0.128217\pi$$
$$30$$ 0 0
$$31$$ −4.21110 −0.756336 −0.378168 0.925737i $$-0.623446\pi$$
−0.378168 + 0.925737i $$0.623446\pi$$
$$32$$ 5.30278 0.937407
$$33$$ 0 0
$$34$$ −9.00000 −1.54349
$$35$$ 0 0
$$36$$ 0 0
$$37$$ 9.60555 1.57914 0.789571 0.613659i $$-0.210303\pi$$
0.789571 + 0.613659i $$0.210303\pi$$
$$38$$ 2.30278 0.373560
$$39$$ 0 0
$$40$$ 0 0
$$41$$ −1.60555 −0.250745 −0.125372 0.992110i $$-0.540013\pi$$
−0.125372 + 0.992110i $$0.540013\pi$$
$$42$$ 0 0
$$43$$ −7.21110 −1.09968 −0.549841 0.835269i $$-0.685312\pi$$
−0.549841 + 0.835269i $$0.685312\pi$$
$$44$$ 3.30278 0.497912
$$45$$ 0 0
$$46$$ −8.51388 −1.25530
$$47$$ 3.00000 0.437595 0.218797 0.975770i $$-0.429787\pi$$
0.218797 + 0.975770i $$0.429787\pi$$
$$48$$ 0 0
$$49$$ 11.5139 1.64484
$$50$$ 0 0
$$51$$ 0 0
$$52$$ 16.5139 2.29006
$$53$$ −2.30278 −0.316311 −0.158155 0.987414i $$-0.550555\pi$$
−0.158155 + 0.987414i $$0.550555\pi$$
$$54$$ 0 0
$$55$$ 0 0
$$56$$ −12.9083 −1.72495
$$57$$ 0 0
$$58$$ −22.8167 −2.99597
$$59$$ −0.211103 −0.0274832 −0.0137416 0.999906i $$-0.504374\pi$$
−0.0137416 + 0.999906i $$0.504374\pi$$
$$60$$ 0 0
$$61$$ 2.90833 0.372373 0.186187 0.982514i $$-0.440387\pi$$
0.186187 + 0.982514i $$0.440387\pi$$
$$62$$ 9.69722 1.23155
$$63$$ 0 0
$$64$$ −12.8167 −1.60208
$$65$$ 0 0
$$66$$ 0 0
$$67$$ −4.00000 −0.488678 −0.244339 0.969690i $$-0.578571\pi$$
−0.244339 + 0.969690i $$0.578571\pi$$
$$68$$ 12.9083 1.56536
$$69$$ 0 0
$$70$$ 0 0
$$71$$ −4.60555 −0.546578 −0.273289 0.961932i $$-0.588112\pi$$
−0.273289 + 0.961932i $$0.588112\pi$$
$$72$$ 0 0
$$73$$ 2.90833 0.340394 0.170197 0.985410i $$-0.445560\pi$$
0.170197 + 0.985410i $$0.445560\pi$$
$$74$$ −22.1194 −2.57133
$$75$$ 0 0
$$76$$ −3.30278 −0.378854
$$77$$ 4.30278 0.490347
$$78$$ 0 0
$$79$$ −0.0916731 −0.0103140 −0.00515701 0.999987i $$-0.501642\pi$$
−0.00515701 + 0.999987i $$0.501642\pi$$
$$80$$ 0 0
$$81$$ 0 0
$$82$$ 3.69722 0.408290
$$83$$ −14.5139 −1.59311 −0.796553 0.604569i $$-0.793345\pi$$
−0.796553 + 0.604569i $$0.793345\pi$$
$$84$$ 0 0
$$85$$ 0 0
$$86$$ 16.6056 1.79062
$$87$$ 0 0
$$88$$ −3.00000 −0.319801
$$89$$ −5.30278 −0.562093 −0.281047 0.959694i $$-0.590682\pi$$
−0.281047 + 0.959694i $$0.590682\pi$$
$$90$$ 0 0
$$91$$ 21.5139 2.25527
$$92$$ 12.2111 1.27310
$$93$$ 0 0
$$94$$ −6.90833 −0.712540
$$95$$ 0 0
$$96$$ 0 0
$$97$$ 11.6972 1.18767 0.593837 0.804586i $$-0.297613\pi$$
0.593837 + 0.804586i $$0.297613\pi$$
$$98$$ −26.5139 −2.67831
$$99$$ 0 0
$$100$$ 0 0
$$101$$ −17.5139 −1.74270 −0.871348 0.490666i $$-0.836754\pi$$
−0.871348 + 0.490666i $$0.836754\pi$$
$$102$$ 0 0
$$103$$ −7.90833 −0.779231 −0.389615 0.920978i $$-0.627392\pi$$
−0.389615 + 0.920978i $$0.627392\pi$$
$$104$$ −15.0000 −1.47087
$$105$$ 0 0
$$106$$ 5.30278 0.515051
$$107$$ 3.00000 0.290021 0.145010 0.989430i $$-0.453678\pi$$
0.145010 + 0.989430i $$0.453678\pi$$
$$108$$ 0 0
$$109$$ −6.51388 −0.623916 −0.311958 0.950096i $$-0.600985\pi$$
−0.311958 + 0.950096i $$0.600985\pi$$
$$110$$ 0 0
$$111$$ 0 0
$$112$$ 1.30278 0.123101
$$113$$ −10.8167 −1.01755 −0.508773 0.860901i $$-0.669901\pi$$
−0.508773 + 0.860901i $$0.669901\pi$$
$$114$$ 0 0
$$115$$ 0 0
$$116$$ 32.7250 3.03844
$$117$$ 0 0
$$118$$ 0.486122 0.0447511
$$119$$ 16.8167 1.54158
$$120$$ 0 0
$$121$$ 1.00000 0.0909091
$$122$$ −6.69722 −0.606338
$$123$$ 0 0
$$124$$ −13.9083 −1.24900
$$125$$ 0 0
$$126$$ 0 0
$$127$$ −17.1194 −1.51910 −0.759552 0.650447i $$-0.774582\pi$$
−0.759552 + 0.650447i $$0.774582\pi$$
$$128$$ 18.9083 1.67128
$$129$$ 0 0
$$130$$ 0 0
$$131$$ −0.908327 −0.0793609 −0.0396804 0.999212i $$-0.512634\pi$$
−0.0396804 + 0.999212i $$0.512634\pi$$
$$132$$ 0 0
$$133$$ −4.30278 −0.373098
$$134$$ 9.21110 0.795718
$$135$$ 0 0
$$136$$ −11.7250 −1.00541
$$137$$ 2.09167 0.178704 0.0893518 0.996000i $$-0.471520\pi$$
0.0893518 + 0.996000i $$0.471520\pi$$
$$138$$ 0 0
$$139$$ 8.21110 0.696457 0.348228 0.937410i $$-0.386783\pi$$
0.348228 + 0.937410i $$0.386783\pi$$
$$140$$ 0 0
$$141$$ 0 0
$$142$$ 10.6056 0.889998
$$143$$ 5.00000 0.418121
$$144$$ 0 0
$$145$$ 0 0
$$146$$ −6.69722 −0.554266
$$147$$ 0 0
$$148$$ 31.7250 2.60778
$$149$$ −2.78890 −0.228475 −0.114238 0.993453i $$-0.536443\pi$$
−0.114238 + 0.993453i $$0.536443\pi$$
$$150$$ 0 0
$$151$$ −20.8167 −1.69404 −0.847018 0.531565i $$-0.821604\pi$$
−0.847018 + 0.531565i $$0.821604\pi$$
$$152$$ 3.00000 0.243332
$$153$$ 0 0
$$154$$ −9.90833 −0.798436
$$155$$ 0 0
$$156$$ 0 0
$$157$$ 4.78890 0.382196 0.191098 0.981571i $$-0.438795\pi$$
0.191098 + 0.981571i $$0.438795\pi$$
$$158$$ 0.211103 0.0167944
$$159$$ 0 0
$$160$$ 0 0
$$161$$ 15.9083 1.25375
$$162$$ 0 0
$$163$$ 5.69722 0.446241 0.223121 0.974791i $$-0.428376\pi$$
0.223121 + 0.974791i $$0.428376\pi$$
$$164$$ −5.30278 −0.414077
$$165$$ 0 0
$$166$$ 33.4222 2.59407
$$167$$ 15.4222 1.19341 0.596703 0.802462i $$-0.296477\pi$$
0.596703 + 0.802462i $$0.296477\pi$$
$$168$$ 0 0
$$169$$ 12.0000 0.923077
$$170$$ 0 0
$$171$$ 0 0
$$172$$ −23.8167 −1.81600
$$173$$ −16.8167 −1.27855 −0.639273 0.768980i $$-0.720765\pi$$
−0.639273 + 0.768980i $$0.720765\pi$$
$$174$$ 0 0
$$175$$ 0 0
$$176$$ 0.302776 0.0228226
$$177$$ 0 0
$$178$$ 12.2111 0.915261
$$179$$ 5.51388 0.412127 0.206063 0.978539i $$-0.433935\pi$$
0.206063 + 0.978539i $$0.433935\pi$$
$$180$$ 0 0
$$181$$ −9.09167 −0.675779 −0.337889 0.941186i $$-0.609713\pi$$
−0.337889 + 0.941186i $$0.609713\pi$$
$$182$$ −49.5416 −3.67227
$$183$$ 0 0
$$184$$ −11.0917 −0.817689
$$185$$ 0 0
$$186$$ 0 0
$$187$$ 3.90833 0.285805
$$188$$ 9.90833 0.722639
$$189$$ 0 0
$$190$$ 0 0
$$191$$ −6.69722 −0.484594 −0.242297 0.970202i $$-0.577901\pi$$
−0.242297 + 0.970202i $$0.577901\pi$$
$$192$$ 0 0
$$193$$ −1.21110 −0.0871771 −0.0435885 0.999050i $$-0.513879\pi$$
−0.0435885 + 0.999050i $$0.513879\pi$$
$$194$$ −26.9361 −1.93390
$$195$$ 0 0
$$196$$ 38.0278 2.71627
$$197$$ −9.69722 −0.690899 −0.345449 0.938437i $$-0.612273\pi$$
−0.345449 + 0.938437i $$0.612273\pi$$
$$198$$ 0 0
$$199$$ −24.5139 −1.73774 −0.868871 0.495038i $$-0.835154\pi$$
−0.868871 + 0.495038i $$0.835154\pi$$
$$200$$ 0 0
$$201$$ 0 0
$$202$$ 40.3305 2.83765
$$203$$ 42.6333 2.99227
$$204$$ 0 0
$$205$$ 0 0
$$206$$ 18.2111 1.26883
$$207$$ 0 0
$$208$$ 1.51388 0.104969
$$209$$ −1.00000 −0.0691714
$$210$$ 0 0
$$211$$ 25.2389 1.73751 0.868757 0.495238i $$-0.164919\pi$$
0.868757 + 0.495238i $$0.164919\pi$$
$$212$$ −7.60555 −0.522351
$$213$$ 0 0
$$214$$ −6.90833 −0.472244
$$215$$ 0 0
$$216$$ 0 0
$$217$$ −18.1194 −1.23003
$$218$$ 15.0000 1.01593
$$219$$ 0 0
$$220$$ 0 0
$$221$$ 19.5416 1.31451
$$222$$ 0 0
$$223$$ 20.6333 1.38171 0.690854 0.722994i $$-0.257235\pi$$
0.690854 + 0.722994i $$0.257235\pi$$
$$224$$ 22.8167 1.52450
$$225$$ 0 0
$$226$$ 24.9083 1.65688
$$227$$ −5.30278 −0.351958 −0.175979 0.984394i $$-0.556309\pi$$
−0.175979 + 0.984394i $$0.556309\pi$$
$$228$$ 0 0
$$229$$ 13.7250 0.906972 0.453486 0.891263i $$-0.350180\pi$$
0.453486 + 0.891263i $$0.350180\pi$$
$$230$$ 0 0
$$231$$ 0 0
$$232$$ −29.7250 −1.95154
$$233$$ −5.09167 −0.333567 −0.166783 0.985994i $$-0.553338\pi$$
−0.166783 + 0.985994i $$0.553338\pi$$
$$234$$ 0 0
$$235$$ 0 0
$$236$$ −0.697224 −0.0453854
$$237$$ 0 0
$$238$$ −38.7250 −2.51017
$$239$$ 4.11943 0.266464 0.133232 0.991085i $$-0.457465\pi$$
0.133232 + 0.991085i $$0.457465\pi$$
$$240$$ 0 0
$$241$$ −24.9361 −1.60627 −0.803137 0.595794i $$-0.796837\pi$$
−0.803137 + 0.595794i $$0.796837\pi$$
$$242$$ −2.30278 −0.148028
$$243$$ 0 0
$$244$$ 9.60555 0.614932
$$245$$ 0 0
$$246$$ 0 0
$$247$$ −5.00000 −0.318142
$$248$$ 12.6333 0.802216
$$249$$ 0 0
$$250$$ 0 0
$$251$$ 3.90833 0.246691 0.123346 0.992364i $$-0.460638\pi$$
0.123346 + 0.992364i $$0.460638\pi$$
$$252$$ 0 0
$$253$$ 3.69722 0.232443
$$254$$ 39.4222 2.47357
$$255$$ 0 0
$$256$$ −17.9083 −1.11927
$$257$$ 18.0000 1.12281 0.561405 0.827541i $$-0.310261\pi$$
0.561405 + 0.827541i $$0.310261\pi$$
$$258$$ 0 0
$$259$$ 41.3305 2.56815
$$260$$ 0 0
$$261$$ 0 0
$$262$$ 2.09167 0.129224
$$263$$ 1.18335 0.0729683 0.0364841 0.999334i $$-0.488384\pi$$
0.0364841 + 0.999334i $$0.488384\pi$$
$$264$$ 0 0
$$265$$ 0 0
$$266$$ 9.90833 0.607519
$$267$$ 0 0
$$268$$ −13.2111 −0.806997
$$269$$ 23.7250 1.44654 0.723269 0.690567i $$-0.242639\pi$$
0.723269 + 0.690567i $$0.242639\pi$$
$$270$$ 0 0
$$271$$ 14.2111 0.863263 0.431632 0.902050i $$-0.357938\pi$$
0.431632 + 0.902050i $$0.357938\pi$$
$$272$$ 1.18335 0.0717509
$$273$$ 0 0
$$274$$ −4.81665 −0.290985
$$275$$ 0 0
$$276$$ 0 0
$$277$$ 21.6056 1.29815 0.649076 0.760724i $$-0.275156\pi$$
0.649076 + 0.760724i $$0.275156\pi$$
$$278$$ −18.9083 −1.13405
$$279$$ 0 0
$$280$$ 0 0
$$281$$ 22.8167 1.36113 0.680564 0.732689i $$-0.261735\pi$$
0.680564 + 0.732689i $$0.261735\pi$$
$$282$$ 0 0
$$283$$ 2.69722 0.160333 0.0801667 0.996781i $$-0.474455\pi$$
0.0801667 + 0.996781i $$0.474455\pi$$
$$284$$ −15.2111 −0.902613
$$285$$ 0 0
$$286$$ −11.5139 −0.680830
$$287$$ −6.90833 −0.407786
$$288$$ 0 0
$$289$$ −1.72498 −0.101469
$$290$$ 0 0
$$291$$ 0 0
$$292$$ 9.60555 0.562122
$$293$$ −15.2111 −0.888642 −0.444321 0.895868i $$-0.646555\pi$$
−0.444321 + 0.895868i $$0.646555\pi$$
$$294$$ 0 0
$$295$$ 0 0
$$296$$ −28.8167 −1.67493
$$297$$ 0 0
$$298$$ 6.42221 0.372028
$$299$$ 18.4861 1.06908
$$300$$ 0 0
$$301$$ −31.0278 −1.78841
$$302$$ 47.9361 2.75841
$$303$$ 0 0
$$304$$ −0.302776 −0.0173654
$$305$$ 0 0
$$306$$ 0 0
$$307$$ −6.09167 −0.347670 −0.173835 0.984775i $$-0.555616\pi$$
−0.173835 + 0.984775i $$0.555616\pi$$
$$308$$ 14.2111 0.809753
$$309$$ 0 0
$$310$$ 0 0
$$311$$ 16.8167 0.953585 0.476792 0.879016i $$-0.341799\pi$$
0.476792 + 0.879016i $$0.341799\pi$$
$$312$$ 0 0
$$313$$ 21.8167 1.23315 0.616575 0.787296i $$-0.288520\pi$$
0.616575 + 0.787296i $$0.288520\pi$$
$$314$$ −11.0278 −0.622332
$$315$$ 0 0
$$316$$ −0.302776 −0.0170325
$$317$$ −9.90833 −0.556507 −0.278254 0.960508i $$-0.589756\pi$$
−0.278254 + 0.960508i $$0.589756\pi$$
$$318$$ 0 0
$$319$$ 9.90833 0.554760
$$320$$ 0 0
$$321$$ 0 0
$$322$$ −36.6333 −2.04149
$$323$$ −3.90833 −0.217465
$$324$$ 0 0
$$325$$ 0 0
$$326$$ −13.1194 −0.726618
$$327$$ 0 0
$$328$$ 4.81665 0.265955
$$329$$ 12.9083 0.711659
$$330$$ 0 0
$$331$$ −14.3944 −0.791190 −0.395595 0.918425i $$-0.629462\pi$$
−0.395595 + 0.918425i $$0.629462\pi$$
$$332$$ −47.9361 −2.63083
$$333$$ 0 0
$$334$$ −35.5139 −1.94323
$$335$$ 0 0
$$336$$ 0 0
$$337$$ 26.8444 1.46231 0.731154 0.682212i $$-0.238982\pi$$
0.731154 + 0.682212i $$0.238982\pi$$
$$338$$ −27.6333 −1.50305
$$339$$ 0 0
$$340$$ 0 0
$$341$$ −4.21110 −0.228044
$$342$$ 0 0
$$343$$ 19.4222 1.04870
$$344$$ 21.6333 1.16639
$$345$$ 0 0
$$346$$ 38.7250 2.08187
$$347$$ −5.51388 −0.296000 −0.148000 0.988987i $$-0.547284\pi$$
−0.148000 + 0.988987i $$0.547284\pi$$
$$348$$ 0 0
$$349$$ −26.8167 −1.43546 −0.717731 0.696320i $$-0.754819\pi$$
−0.717731 + 0.696320i $$0.754819\pi$$
$$350$$ 0 0
$$351$$ 0 0
$$352$$ 5.30278 0.282639
$$353$$ 24.6333 1.31110 0.655549 0.755152i $$-0.272437\pi$$
0.655549 + 0.755152i $$0.272437\pi$$
$$354$$ 0 0
$$355$$ 0 0
$$356$$ −17.5139 −0.928234
$$357$$ 0 0
$$358$$ −12.6972 −0.671069
$$359$$ −15.2111 −0.802811 −0.401406 0.915900i $$-0.631478\pi$$
−0.401406 + 0.915900i $$0.631478\pi$$
$$360$$ 0 0
$$361$$ −18.0000 −0.947368
$$362$$ 20.9361 1.10038
$$363$$ 0 0
$$364$$ 71.0555 3.72432
$$365$$ 0 0
$$366$$ 0 0
$$367$$ −24.3028 −1.26859 −0.634297 0.773089i $$-0.718710\pi$$
−0.634297 + 0.773089i $$0.718710\pi$$
$$368$$ 1.11943 0.0583543
$$369$$ 0 0
$$370$$ 0 0
$$371$$ −9.90833 −0.514415
$$372$$ 0 0
$$373$$ −1.42221 −0.0736390 −0.0368195 0.999322i $$-0.511723\pi$$
−0.0368195 + 0.999322i $$0.511723\pi$$
$$374$$ −9.00000 −0.465379
$$375$$ 0 0
$$376$$ −9.00000 −0.464140
$$377$$ 49.5416 2.55152
$$378$$ 0 0
$$379$$ 24.8167 1.27475 0.637373 0.770555i $$-0.280021\pi$$
0.637373 + 0.770555i $$0.280021\pi$$
$$380$$ 0 0
$$381$$ 0 0
$$382$$ 15.4222 0.789069
$$383$$ 21.6333 1.10541 0.552705 0.833377i $$-0.313596\pi$$
0.552705 + 0.833377i $$0.313596\pi$$
$$384$$ 0 0
$$385$$ 0 0
$$386$$ 2.78890 0.141951
$$387$$ 0 0
$$388$$ 38.6333 1.96131
$$389$$ −12.0000 −0.608424 −0.304212 0.952604i $$-0.598393\pi$$
−0.304212 + 0.952604i $$0.598393\pi$$
$$390$$ 0 0
$$391$$ 14.4500 0.730766
$$392$$ −34.5416 −1.74462
$$393$$ 0 0
$$394$$ 22.3305 1.12500
$$395$$ 0 0
$$396$$ 0 0
$$397$$ 25.3028 1.26991 0.634955 0.772549i $$-0.281019\pi$$
0.634955 + 0.772549i $$0.281019\pi$$
$$398$$ 56.4500 2.82958
$$399$$ 0 0
$$400$$ 0 0
$$401$$ 27.2111 1.35886 0.679429 0.733741i $$-0.262228\pi$$
0.679429 + 0.733741i $$0.262228\pi$$
$$402$$ 0 0
$$403$$ −21.0555 −1.04885
$$404$$ −57.8444 −2.87787
$$405$$ 0 0
$$406$$ −98.1749 −4.87234
$$407$$ 9.60555 0.476129
$$408$$ 0 0
$$409$$ 8.21110 0.406013 0.203006 0.979177i $$-0.434929\pi$$
0.203006 + 0.979177i $$0.434929\pi$$
$$410$$ 0 0
$$411$$ 0 0
$$412$$ −26.1194 −1.28681
$$413$$ −0.908327 −0.0446958
$$414$$ 0 0
$$415$$ 0 0
$$416$$ 26.5139 1.29995
$$417$$ 0 0
$$418$$ 2.30278 0.112632
$$419$$ −13.6056 −0.664675 −0.332337 0.943161i $$-0.607837\pi$$
−0.332337 + 0.943161i $$0.607837\pi$$
$$420$$ 0 0
$$421$$ 4.30278 0.209704 0.104852 0.994488i $$-0.466563\pi$$
0.104852 + 0.994488i $$0.466563\pi$$
$$422$$ −58.1194 −2.82921
$$423$$ 0 0
$$424$$ 6.90833 0.335498
$$425$$ 0 0
$$426$$ 0 0
$$427$$ 12.5139 0.605589
$$428$$ 9.90833 0.478937
$$429$$ 0 0
$$430$$ 0 0
$$431$$ −33.0000 −1.58955 −0.794777 0.606902i $$-0.792412\pi$$
−0.794777 + 0.606902i $$0.792412\pi$$
$$432$$ 0 0
$$433$$ 5.00000 0.240285 0.120142 0.992757i $$-0.461665\pi$$
0.120142 + 0.992757i $$0.461665\pi$$
$$434$$ 41.7250 2.00286
$$435$$ 0 0
$$436$$ −21.5139 −1.03033
$$437$$ −3.69722 −0.176862
$$438$$ 0 0
$$439$$ 20.6972 0.987825 0.493912 0.869512i $$-0.335566\pi$$
0.493912 + 0.869512i $$0.335566\pi$$
$$440$$ 0 0
$$441$$ 0 0
$$442$$ −45.0000 −2.14043
$$443$$ 1.39445 0.0662523 0.0331261 0.999451i $$-0.489454\pi$$
0.0331261 + 0.999451i $$0.489454\pi$$
$$444$$ 0 0
$$445$$ 0 0
$$446$$ −47.5139 −2.24985
$$447$$ 0 0
$$448$$ −55.1472 −2.60546
$$449$$ −41.5139 −1.95916 −0.979581 0.201052i $$-0.935564\pi$$
−0.979581 + 0.201052i $$0.935564\pi$$
$$450$$ 0 0
$$451$$ −1.60555 −0.0756025
$$452$$ −35.7250 −1.68036
$$453$$ 0 0
$$454$$ 12.2111 0.573095
$$455$$ 0 0
$$456$$ 0 0
$$457$$ −24.3028 −1.13684 −0.568418 0.822740i $$-0.692444\pi$$
−0.568418 + 0.822740i $$0.692444\pi$$
$$458$$ −31.6056 −1.47683
$$459$$ 0 0
$$460$$ 0 0
$$461$$ −17.7889 −0.828512 −0.414256 0.910161i $$-0.635958\pi$$
−0.414256 + 0.910161i $$0.635958\pi$$
$$462$$ 0 0
$$463$$ 26.2111 1.21813 0.609067 0.793119i $$-0.291544\pi$$
0.609067 + 0.793119i $$0.291544\pi$$
$$464$$ 3.00000 0.139272
$$465$$ 0 0
$$466$$ 11.7250 0.543149
$$467$$ −24.6333 −1.13989 −0.569947 0.821682i $$-0.693036\pi$$
−0.569947 + 0.821682i $$0.693036\pi$$
$$468$$ 0 0
$$469$$ −17.2111 −0.794735
$$470$$ 0 0
$$471$$ 0 0
$$472$$ 0.633308 0.0291503
$$473$$ −7.21110 −0.331567
$$474$$ 0 0
$$475$$ 0 0
$$476$$ 55.5416 2.54575
$$477$$ 0 0
$$478$$ −9.48612 −0.433885
$$479$$ 13.1833 0.602362 0.301181 0.953567i $$-0.402619\pi$$
0.301181 + 0.953567i $$0.402619\pi$$
$$480$$ 0 0
$$481$$ 48.0278 2.18988
$$482$$ 57.4222 2.61551
$$483$$ 0 0
$$484$$ 3.30278 0.150126
$$485$$ 0 0
$$486$$ 0 0
$$487$$ −10.2111 −0.462709 −0.231355 0.972869i $$-0.574316\pi$$
−0.231355 + 0.972869i $$0.574316\pi$$
$$488$$ −8.72498 −0.394961
$$489$$ 0 0
$$490$$ 0 0
$$491$$ 24.2111 1.09263 0.546316 0.837579i $$-0.316030\pi$$
0.546316 + 0.837579i $$0.316030\pi$$
$$492$$ 0 0
$$493$$ 38.7250 1.74409
$$494$$ 11.5139 0.518034
$$495$$ 0 0
$$496$$ −1.27502 −0.0572501
$$497$$ −19.8167 −0.888898
$$498$$ 0 0
$$499$$ −21.5139 −0.963093 −0.481547 0.876420i $$-0.659925\pi$$
−0.481547 + 0.876420i $$0.659925\pi$$
$$500$$ 0 0
$$501$$ 0 0
$$502$$ −9.00000 −0.401690
$$503$$ −16.6056 −0.740405 −0.370202 0.928951i $$-0.620712\pi$$
−0.370202 + 0.928951i $$0.620712\pi$$
$$504$$ 0 0
$$505$$ 0 0
$$506$$ −8.51388 −0.378488
$$507$$ 0 0
$$508$$ −56.5416 −2.50863
$$509$$ 26.3028 1.16585 0.582925 0.812526i $$-0.301908\pi$$
0.582925 + 0.812526i $$0.301908\pi$$
$$510$$ 0 0
$$511$$ 12.5139 0.553581
$$512$$ 3.42221 0.151242
$$513$$ 0 0
$$514$$ −41.4500 −1.82828
$$515$$ 0 0
$$516$$ 0 0
$$517$$ 3.00000 0.131940
$$518$$ −95.1749 −4.18175
$$519$$ 0 0
$$520$$ 0 0
$$521$$ −23.4500 −1.02736 −0.513681 0.857981i $$-0.671718\pi$$
−0.513681 + 0.857981i $$0.671718\pi$$
$$522$$ 0 0
$$523$$ −3.57779 −0.156446 −0.0782230 0.996936i $$-0.524925\pi$$
−0.0782230 + 0.996936i $$0.524925\pi$$
$$524$$ −3.00000 −0.131056
$$525$$ 0 0
$$526$$ −2.72498 −0.118815
$$527$$ −16.4584 −0.716938
$$528$$ 0 0
$$529$$ −9.33053 −0.405675
$$530$$ 0 0
$$531$$ 0 0
$$532$$ −14.2111 −0.616129
$$533$$ −8.02776 −0.347721
$$534$$ 0 0
$$535$$ 0 0
$$536$$ 12.0000 0.518321
$$537$$ 0 0
$$538$$ −54.6333 −2.35541
$$539$$ 11.5139 0.495938
$$540$$ 0 0
$$541$$ −6.72498 −0.289130 −0.144565 0.989495i $$-0.546178\pi$$
−0.144565 + 0.989495i $$0.546178\pi$$
$$542$$ −32.7250 −1.40566
$$543$$ 0 0
$$544$$ 20.7250 0.888576
$$545$$ 0 0
$$546$$ 0 0
$$547$$ 18.1194 0.774731 0.387365 0.921926i $$-0.373385\pi$$
0.387365 + 0.921926i $$0.373385\pi$$
$$548$$ 6.90833 0.295109
$$549$$ 0 0
$$550$$ 0 0
$$551$$ −9.90833 −0.422109
$$552$$ 0 0
$$553$$ −0.394449 −0.0167737
$$554$$ −49.7527 −2.11379
$$555$$ 0 0
$$556$$ 27.1194 1.15012
$$557$$ 9.42221 0.399232 0.199616 0.979874i $$-0.436031\pi$$
0.199616 + 0.979874i $$0.436031\pi$$
$$558$$ 0 0
$$559$$ −36.0555 −1.52499
$$560$$ 0 0
$$561$$ 0 0
$$562$$ −52.5416 −2.21634
$$563$$ −18.9083 −0.796891 −0.398445 0.917192i $$-0.630450\pi$$
−0.398445 + 0.917192i $$0.630450\pi$$
$$564$$ 0 0
$$565$$ 0 0
$$566$$ −6.21110 −0.261072
$$567$$ 0 0
$$568$$ 13.8167 0.579734
$$569$$ −15.1472 −0.635003 −0.317502 0.948258i $$-0.602844\pi$$
−0.317502 + 0.948258i $$0.602844\pi$$
$$570$$ 0 0
$$571$$ −17.3305 −0.725260 −0.362630 0.931933i $$-0.618121\pi$$
−0.362630 + 0.931933i $$0.618121\pi$$
$$572$$ 16.5139 0.690480
$$573$$ 0 0
$$574$$ 15.9083 0.664001
$$575$$ 0 0
$$576$$ 0 0
$$577$$ −31.3583 −1.30546 −0.652731 0.757589i $$-0.726377\pi$$
−0.652731 + 0.757589i $$0.726377\pi$$
$$578$$ 3.97224 0.165224
$$579$$ 0 0
$$580$$ 0 0
$$581$$ −62.4500 −2.59086
$$582$$ 0 0
$$583$$ −2.30278 −0.0953712
$$584$$ −8.72498 −0.361042
$$585$$ 0 0
$$586$$ 35.0278 1.44698
$$587$$ 37.5416 1.54951 0.774755 0.632262i $$-0.217873\pi$$
0.774755 + 0.632262i $$0.217873\pi$$
$$588$$ 0 0
$$589$$ 4.21110 0.173515
$$590$$ 0 0
$$591$$ 0 0
$$592$$ 2.90833 0.119531
$$593$$ 13.6056 0.558713 0.279357 0.960187i $$-0.409879\pi$$
0.279357 + 0.960187i $$0.409879\pi$$
$$594$$ 0 0
$$595$$ 0 0
$$596$$ −9.21110 −0.377301
$$597$$ 0 0
$$598$$ −42.5694 −1.74079
$$599$$ 14.0917 0.575770 0.287885 0.957665i $$-0.407048\pi$$
0.287885 + 0.957665i $$0.407048\pi$$
$$600$$ 0 0
$$601$$ 8.90833 0.363378 0.181689 0.983356i $$-0.441844\pi$$
0.181689 + 0.983356i $$0.441844\pi$$
$$602$$ 71.4500 2.91208
$$603$$ 0 0
$$604$$ −68.7527 −2.79751
$$605$$ 0 0
$$606$$ 0 0
$$607$$ −7.21110 −0.292690 −0.146345 0.989234i $$-0.546751\pi$$
−0.146345 + 0.989234i $$0.546751\pi$$
$$608$$ −5.30278 −0.215056
$$609$$ 0 0
$$610$$ 0 0
$$611$$ 15.0000 0.606835
$$612$$ 0 0
$$613$$ −41.1194 −1.66080 −0.830399 0.557169i $$-0.811888\pi$$
−0.830399 + 0.557169i $$0.811888\pi$$
$$614$$ 14.0278 0.566114
$$615$$ 0 0
$$616$$ −12.9083 −0.520091
$$617$$ −10.6056 −0.426963 −0.213482 0.976947i $$-0.568480\pi$$
−0.213482 + 0.976947i $$0.568480\pi$$
$$618$$ 0 0
$$619$$ 17.4222 0.700258 0.350129 0.936702i $$-0.386138\pi$$
0.350129 + 0.936702i $$0.386138\pi$$
$$620$$ 0 0
$$621$$ 0 0
$$622$$ −38.7250 −1.55273
$$623$$ −22.8167 −0.914130
$$624$$ 0 0
$$625$$ 0 0
$$626$$ −50.2389 −2.00795
$$627$$ 0 0
$$628$$ 15.8167 0.631153
$$629$$ 37.5416 1.49688
$$630$$ 0 0
$$631$$ −39.9361 −1.58983 −0.794915 0.606721i $$-0.792485\pi$$
−0.794915 + 0.606721i $$0.792485\pi$$
$$632$$ 0.275019 0.0109397
$$633$$ 0 0
$$634$$ 22.8167 0.906165
$$635$$ 0 0
$$636$$ 0 0
$$637$$ 57.5694 2.28098
$$638$$ −22.8167 −0.903320
$$639$$ 0 0
$$640$$ 0 0
$$641$$ −42.2111 −1.66724 −0.833619 0.552340i $$-0.813735\pi$$
−0.833619 + 0.552340i $$0.813735\pi$$
$$642$$ 0 0
$$643$$ −22.0000 −0.867595 −0.433798 0.901010i $$-0.642827\pi$$
−0.433798 + 0.901010i $$0.642827\pi$$
$$644$$ 52.5416 2.07043
$$645$$ 0 0
$$646$$ 9.00000 0.354100
$$647$$ −17.2389 −0.677729 −0.338865 0.940835i $$-0.610043\pi$$
−0.338865 + 0.940835i $$0.610043\pi$$
$$648$$ 0 0
$$649$$ −0.211103 −0.00828650
$$650$$ 0 0
$$651$$ 0 0
$$652$$ 18.8167 0.736917
$$653$$ 19.1194 0.748201 0.374101 0.927388i $$-0.377951\pi$$
0.374101 + 0.927388i $$0.377951\pi$$
$$654$$ 0 0
$$655$$ 0 0
$$656$$ −0.486122 −0.0189799
$$657$$ 0 0
$$658$$ −29.7250 −1.15880
$$659$$ 20.0917 0.782660 0.391330 0.920250i $$-0.372015\pi$$
0.391330 + 0.920250i $$0.372015\pi$$
$$660$$ 0 0
$$661$$ 12.8167 0.498510 0.249255 0.968438i $$-0.419814\pi$$
0.249255 + 0.968438i $$0.419814\pi$$
$$662$$ 33.1472 1.28830
$$663$$ 0 0
$$664$$ 43.5416 1.68974
$$665$$ 0 0
$$666$$ 0 0
$$667$$ 36.6333 1.41845
$$668$$ 50.9361 1.97078
$$669$$ 0 0
$$670$$ 0 0
$$671$$ 2.90833 0.112275
$$672$$ 0 0
$$673$$ −6.02776 −0.232353 −0.116176 0.993229i $$-0.537064\pi$$
−0.116176 + 0.993229i $$0.537064\pi$$
$$674$$ −61.8167 −2.38109
$$675$$ 0 0
$$676$$ 39.6333 1.52436
$$677$$ 26.2389 1.00844 0.504221 0.863575i $$-0.331780\pi$$
0.504221 + 0.863575i $$0.331780\pi$$
$$678$$ 0 0
$$679$$ 50.3305 1.93151
$$680$$ 0 0
$$681$$ 0 0
$$682$$ 9.69722 0.371326
$$683$$ 9.84441 0.376686 0.188343 0.982103i $$-0.439688\pi$$
0.188343 + 0.982103i $$0.439688\pi$$
$$684$$ 0 0
$$685$$ 0 0
$$686$$ −44.7250 −1.70761
$$687$$ 0 0
$$688$$ −2.18335 −0.0832393
$$689$$ −11.5139 −0.438644
$$690$$ 0 0
$$691$$ −26.5416 −1.00969 −0.504846 0.863210i $$-0.668451\pi$$
−0.504846 + 0.863210i $$0.668451\pi$$
$$692$$ −55.5416 −2.11138
$$693$$ 0 0
$$694$$ 12.6972 0.481980
$$695$$ 0 0
$$696$$ 0 0
$$697$$ −6.27502 −0.237683
$$698$$ 61.7527 2.33738
$$699$$ 0 0
$$700$$ 0 0
$$701$$ 26.7889 1.01180 0.505901 0.862591i $$-0.331160\pi$$
0.505901 + 0.862591i $$0.331160\pi$$
$$702$$ 0 0
$$703$$ −9.60555 −0.362280
$$704$$ −12.8167 −0.483046
$$705$$ 0 0
$$706$$ −56.7250 −2.13487
$$707$$ −75.3583 −2.83414
$$708$$ 0 0
$$709$$ 11.6333 0.436898 0.218449 0.975848i $$-0.429900\pi$$
0.218449 + 0.975848i $$0.429900\pi$$
$$710$$ 0 0
$$711$$ 0 0
$$712$$ 15.9083 0.596190
$$713$$ −15.5694 −0.583078
$$714$$ 0 0
$$715$$ 0 0
$$716$$ 18.2111 0.680581
$$717$$ 0 0
$$718$$ 35.0278 1.30722
$$719$$ −28.8167 −1.07468 −0.537340 0.843366i $$-0.680571\pi$$
−0.537340 + 0.843366i $$0.680571\pi$$
$$720$$ 0 0
$$721$$ −34.0278 −1.26726
$$722$$ 41.4500 1.54261
$$723$$ 0 0
$$724$$ −30.0278 −1.11597
$$725$$ 0 0
$$726$$ 0 0
$$727$$ 0.330532 0.0122588 0.00612938 0.999981i $$-0.498049\pi$$
0.00612938 + 0.999981i $$0.498049\pi$$
$$728$$ −64.5416 −2.39207
$$729$$ 0 0
$$730$$ 0 0
$$731$$ −28.1833 −1.04240
$$732$$ 0 0
$$733$$ 12.3944 0.457799 0.228900 0.973450i $$-0.426487\pi$$
0.228900 + 0.973450i $$0.426487\pi$$
$$734$$ 55.9638 2.06566
$$735$$ 0 0
$$736$$ 19.6056 0.722670
$$737$$ −4.00000 −0.147342
$$738$$ 0 0
$$739$$ 9.88057 0.363463 0.181731 0.983348i $$-0.441830\pi$$
0.181731 + 0.983348i $$0.441830\pi$$
$$740$$ 0 0
$$741$$ 0 0
$$742$$ 22.8167 0.837626
$$743$$ −44.3028 −1.62531 −0.812656 0.582744i $$-0.801979\pi$$
−0.812656 + 0.582744i $$0.801979\pi$$
$$744$$ 0 0
$$745$$ 0 0
$$746$$ 3.27502 0.119907
$$747$$ 0 0
$$748$$ 12.9083 0.471975
$$749$$ 12.9083 0.471660
$$750$$ 0 0
$$751$$ −5.66947 −0.206882 −0.103441 0.994636i $$-0.532985\pi$$
−0.103441 + 0.994636i $$0.532985\pi$$
$$752$$ 0.908327 0.0331233
$$753$$ 0 0
$$754$$ −114.083 −4.15467
$$755$$ 0 0
$$756$$ 0 0
$$757$$ −23.0555 −0.837967 −0.418983 0.907994i $$-0.637613\pi$$
−0.418983 + 0.907994i $$0.637613\pi$$
$$758$$ −57.1472 −2.07568
$$759$$ 0 0
$$760$$ 0 0
$$761$$ 42.4222 1.53780 0.768902 0.639367i $$-0.220803\pi$$
0.768902 + 0.639367i $$0.220803\pi$$
$$762$$ 0 0
$$763$$ −28.0278 −1.01467
$$764$$ −22.1194 −0.800253
$$765$$ 0 0
$$766$$ −49.8167 −1.79995
$$767$$ −1.05551 −0.0381124
$$768$$ 0 0
$$769$$ −26.8167 −0.967033 −0.483517 0.875335i $$-0.660641\pi$$
−0.483517 + 0.875335i $$0.660641\pi$$
$$770$$ 0 0
$$771$$ 0 0
$$772$$ −4.00000 −0.143963
$$773$$ 22.1194 0.795581 0.397790 0.917476i $$-0.369777\pi$$
0.397790 + 0.917476i $$0.369777\pi$$
$$774$$ 0 0
$$775$$ 0 0
$$776$$ −35.0917 −1.25972
$$777$$ 0 0
$$778$$ 27.6333 0.990702
$$779$$ 1.60555 0.0575248
$$780$$ 0 0
$$781$$ −4.60555 −0.164800
$$782$$ −33.2750 −1.18991
$$783$$ 0 0
$$784$$ 3.48612 0.124504
$$785$$ 0 0
$$786$$ 0 0
$$787$$ −4.21110 −0.150110 −0.0750548 0.997179i $$-0.523913\pi$$
−0.0750548 + 0.997179i $$0.523913\pi$$
$$788$$ −32.0278 −1.14094
$$789$$ 0 0
$$790$$ 0 0
$$791$$ −46.5416 −1.65483
$$792$$ 0 0
$$793$$ 14.5416 0.516389
$$794$$ −58.2666 −2.06780
$$795$$ 0 0
$$796$$ −80.9638 −2.86969
$$797$$ 14.5139 0.514108 0.257054 0.966397i $$-0.417248\pi$$
0.257054 + 0.966397i $$0.417248\pi$$
$$798$$ 0 0
$$799$$ 11.7250 0.414800
$$800$$ 0 0
$$801$$ 0 0
$$802$$ −62.6611 −2.21264
$$803$$ 2.90833 0.102633
$$804$$ 0 0
$$805$$ 0 0
$$806$$ 48.4861 1.70785
$$807$$ 0 0
$$808$$ 52.5416 1.84841
$$809$$ 3.63331 0.127740 0.0638701 0.997958i $$-0.479656\pi$$
0.0638701 + 0.997958i $$0.479656\pi$$
$$810$$ 0 0
$$811$$ −54.8722 −1.92682 −0.963411 0.268028i $$-0.913628\pi$$
−0.963411 + 0.268028i $$0.913628\pi$$
$$812$$ 140.808 4.94140
$$813$$ 0 0
$$814$$ −22.1194 −0.775286
$$815$$ 0 0
$$816$$ 0 0
$$817$$ 7.21110 0.252285
$$818$$ −18.9083 −0.661114
$$819$$ 0 0
$$820$$ 0 0
$$821$$ −12.0000 −0.418803 −0.209401 0.977830i $$-0.567152\pi$$
−0.209401 + 0.977830i $$0.567152\pi$$
$$822$$ 0 0
$$823$$ −10.4222 −0.363295 −0.181648 0.983364i $$-0.558143\pi$$
−0.181648 + 0.983364i $$0.558143\pi$$
$$824$$ 23.7250 0.826499
$$825$$ 0 0
$$826$$ 2.09167 0.0727786
$$827$$ 7.81665 0.271812 0.135906 0.990722i $$-0.456606\pi$$
0.135906 + 0.990722i $$0.456606\pi$$
$$828$$ 0 0
$$829$$ −38.7527 −1.34594 −0.672969 0.739671i $$-0.734981\pi$$
−0.672969 + 0.739671i $$0.734981\pi$$
$$830$$ 0 0
$$831$$ 0 0
$$832$$ −64.0833 −2.22169
$$833$$ 45.0000 1.55916
$$834$$ 0 0
$$835$$ 0 0
$$836$$ −3.30278 −0.114229
$$837$$ 0 0
$$838$$ 31.3305 1.08230
$$839$$ −16.1194 −0.556505 −0.278252 0.960508i $$-0.589755\pi$$
−0.278252 + 0.960508i $$0.589755\pi$$
$$840$$ 0 0
$$841$$ 69.1749 2.38534
$$842$$ −9.90833 −0.341463
$$843$$ 0 0
$$844$$ 83.3583 2.86931
$$845$$ 0 0
$$846$$ 0 0
$$847$$ 4.30278 0.147845
$$848$$ −0.697224 −0.0239428
$$849$$ 0 0
$$850$$ 0 0
$$851$$ 35.5139 1.21740
$$852$$ 0 0
$$853$$ 19.7250 0.675370 0.337685 0.941259i $$-0.390356\pi$$
0.337685 + 0.941259i $$0.390356\pi$$
$$854$$ −28.8167 −0.986086
$$855$$ 0 0
$$856$$ −9.00000 −0.307614
$$857$$ 3.00000 0.102478 0.0512390 0.998686i $$-0.483683\pi$$
0.0512390 + 0.998686i $$0.483683\pi$$
$$858$$ 0 0
$$859$$ 48.6056 1.65840 0.829200 0.558952i $$-0.188796\pi$$
0.829200 + 0.558952i $$0.188796\pi$$
$$860$$ 0 0
$$861$$ 0 0
$$862$$ 75.9916 2.58828
$$863$$ 19.6056 0.667381 0.333690 0.942683i $$-0.391706\pi$$
0.333690 + 0.942683i $$0.391706\pi$$
$$864$$ 0 0
$$865$$ 0 0
$$866$$ −11.5139 −0.391258
$$867$$ 0 0
$$868$$ −59.8444 −2.03125
$$869$$ −0.0916731 −0.00310980
$$870$$ 0 0
$$871$$ −20.0000 −0.677674
$$872$$ 19.5416 0.661763
$$873$$ 0 0
$$874$$ 8.51388 0.287986
$$875$$ 0 0
$$876$$ 0 0
$$877$$ 20.0000 0.675352 0.337676 0.941262i $$-0.390359\pi$$
0.337676 + 0.941262i $$0.390359\pi$$
$$878$$ −47.6611 −1.60848
$$879$$ 0 0
$$880$$ 0 0
$$881$$ 34.5416 1.16374 0.581869 0.813283i $$-0.302322\pi$$
0.581869 + 0.813283i $$0.302322\pi$$
$$882$$ 0 0
$$883$$ −12.4500 −0.418975 −0.209487 0.977811i $$-0.567179\pi$$
−0.209487 + 0.977811i $$0.567179\pi$$
$$884$$ 64.5416 2.17077
$$885$$ 0 0
$$886$$ −3.21110 −0.107879
$$887$$ −47.2389 −1.58613 −0.793063 0.609140i $$-0.791515\pi$$
−0.793063 + 0.609140i $$0.791515\pi$$
$$888$$ 0 0
$$889$$ −73.6611 −2.47051
$$890$$ 0 0
$$891$$ 0 0
$$892$$ 68.1472 2.28174
$$893$$ −3.00000 −0.100391
$$894$$ 0 0
$$895$$ 0 0
$$896$$ 81.3583 2.71799
$$897$$ 0 0
$$898$$ 95.5971 3.19012
$$899$$ −41.7250 −1.39161
$$900$$ 0 0
$$901$$ −9.00000 −0.299833
$$902$$ 3.69722 0.123104
$$903$$ 0 0
$$904$$ 32.4500 1.07927
$$905$$ 0 0
$$906$$ 0 0
$$907$$ −4.00000 −0.132818 −0.0664089 0.997792i $$-0.521154\pi$$
−0.0664089 + 0.997792i $$0.521154\pi$$
$$908$$ −17.5139 −0.581218
$$909$$ 0 0
$$910$$ 0 0
$$911$$ 39.2111 1.29912 0.649561 0.760310i $$-0.274953\pi$$
0.649561 + 0.760310i $$0.274953\pi$$
$$912$$ 0 0
$$913$$ −14.5139 −0.480339
$$914$$ 55.9638 1.85112
$$915$$ 0 0
$$916$$ 45.3305 1.49776
$$917$$ −3.90833 −0.129064
$$918$$ 0 0
$$919$$ 41.2111 1.35943 0.679714 0.733477i $$-0.262104\pi$$
0.679714 + 0.733477i $$0.262104\pi$$
$$920$$ 0 0
$$921$$ 0 0
$$922$$ 40.9638 1.34907
$$923$$ −23.0278 −0.757968
$$924$$ 0 0
$$925$$ 0 0
$$926$$ −60.3583 −1.98350
$$927$$ 0 0
$$928$$ 52.5416 1.72476
$$929$$ −46.3944 −1.52215 −0.761076 0.648662i $$-0.775329\pi$$
−0.761076 + 0.648662i $$0.775329\pi$$
$$930$$ 0 0
$$931$$ −11.5139 −0.377352
$$932$$ −16.8167 −0.550848
$$933$$ 0 0
$$934$$ 56.7250 1.85610
$$935$$ 0 0
$$936$$ 0 0
$$937$$ 5.21110 0.170239 0.0851196 0.996371i $$-0.472873\pi$$
0.0851196 + 0.996371i $$0.472873\pi$$
$$938$$ 39.6333 1.29407
$$939$$ 0 0
$$940$$ 0 0
$$941$$ −52.3944 −1.70801 −0.854005 0.520265i $$-0.825833\pi$$
−0.854005 + 0.520265i $$0.825833\pi$$
$$942$$ 0 0
$$943$$ −5.93608 −0.193305
$$944$$ −0.0639167 −0.00208031
$$945$$ 0 0
$$946$$ 16.6056 0.539893
$$947$$ 36.6333 1.19042 0.595211 0.803569i $$-0.297068\pi$$
0.595211 + 0.803569i $$0.297068\pi$$
$$948$$ 0 0
$$949$$ 14.5416 0.472041
$$950$$ 0 0
$$951$$ 0 0
$$952$$ −50.4500 −1.63509
$$953$$ 49.2666 1.59590 0.797951 0.602722i $$-0.205917\pi$$
0.797951 + 0.602722i $$0.205917\pi$$
$$954$$ 0 0
$$955$$ 0 0
$$956$$ 13.6056 0.440035
$$957$$ 0 0
$$958$$ −30.3583 −0.980832
$$959$$ 9.00000 0.290625
$$960$$ 0 0
$$961$$ −13.2666 −0.427955
$$962$$ −110.597 −3.56580
$$963$$ 0 0
$$964$$ −82.3583 −2.65258
$$965$$ 0 0
$$966$$ 0 0
$$967$$ 4.09167 0.131579 0.0657897 0.997834i $$-0.479043\pi$$
0.0657897 + 0.997834i $$0.479043\pi$$
$$968$$ −3.00000 −0.0964237
$$969$$ 0 0
$$970$$ 0 0
$$971$$ 30.3583 0.974244 0.487122 0.873334i $$-0.338047\pi$$
0.487122 + 0.873334i $$0.338047\pi$$
$$972$$ 0 0
$$973$$ 35.3305 1.13264
$$974$$ 23.5139 0.753433
$$975$$ 0 0
$$976$$ 0.880571 0.0281864
$$977$$ 15.9722 0.510997 0.255499 0.966809i $$-0.417760\pi$$
0.255499 + 0.966809i $$0.417760\pi$$
$$978$$ 0 0
$$979$$ −5.30278 −0.169477
$$980$$ 0 0
$$981$$ 0 0
$$982$$ −55.7527 −1.77914
$$983$$ 48.8444 1.55789 0.778947 0.627089i $$-0.215754\pi$$
0.778947 + 0.627089i $$0.215754\pi$$
$$984$$ 0 0
$$985$$ 0 0
$$986$$ −89.1749 −2.83991
$$987$$ 0 0
$$988$$ −16.5139 −0.525376
$$989$$ −26.6611 −0.847773
$$990$$ 0 0
$$991$$ −6.09167 −0.193508 −0.0967542 0.995308i $$-0.530846\pi$$
−0.0967542 + 0.995308i $$0.530846\pi$$
$$992$$ −22.3305 −0.708995
$$993$$ 0 0
$$994$$ 45.6333 1.44740
$$995$$ 0 0
$$996$$ 0 0
$$997$$ 14.2750 0.452094 0.226047 0.974116i $$-0.427420\pi$$
0.226047 + 0.974116i $$0.427420\pi$$
$$998$$ 49.5416 1.56821
$$999$$ 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 2475.2.a.o.1.1 2
3.2 odd 2 275.2.a.f.1.2 yes 2
5.2 odd 4 2475.2.c.k.199.1 4
5.3 odd 4 2475.2.c.k.199.4 4
5.4 even 2 2475.2.a.t.1.2 2
12.11 even 2 4400.2.a.bh.1.2 2
15.2 even 4 275.2.b.c.199.4 4
15.8 even 4 275.2.b.c.199.1 4
15.14 odd 2 275.2.a.e.1.1 2
33.32 even 2 3025.2.a.h.1.1 2
60.23 odd 4 4400.2.b.y.4049.3 4
60.47 odd 4 4400.2.b.y.4049.2 4
60.59 even 2 4400.2.a.bs.1.1 2
165.164 even 2 3025.2.a.n.1.2 2

By twisted newform
Twist Min Dim Char Parity Ord Type
275.2.a.e.1.1 2 15.14 odd 2
275.2.a.f.1.2 yes 2 3.2 odd 2
275.2.b.c.199.1 4 15.8 even 4
275.2.b.c.199.4 4 15.2 even 4
2475.2.a.o.1.1 2 1.1 even 1 trivial
2475.2.a.t.1.2 2 5.4 even 2
2475.2.c.k.199.1 4 5.2 odd 4
2475.2.c.k.199.4 4 5.3 odd 4
3025.2.a.h.1.1 2 33.32 even 2
3025.2.a.n.1.2 2 165.164 even 2
4400.2.a.bh.1.2 2 12.11 even 2
4400.2.a.bs.1.1 2 60.59 even 2
4400.2.b.y.4049.2 4 60.47 odd 4
4400.2.b.y.4049.3 4 60.23 odd 4